Results of Bernoulli and Euler Polynomials
| DLMF | Formula | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|
| 24.2.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{t}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullinumberB{n}\frac{t^{n}}{n!}} | (t)/(exp(t)- 1)= sum(bernoulli(n)*((t)^(n))/(factorial(n)), n = 0..infinity) |
Divide[t,Exp[t]- 1]= Sum[BernoulliB[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 24.2#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n+1} = 0} | bernoulli(2*n + 1)= 0 |
BernoulliB[2*n + 1]= 0 |
Failure | Failure | Successful | Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 24.2#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n+1}\BernoullinumberB{2n} > 0} | (- 1)^(n + 1)* bernoulli(2*n)> 0 |
(- 1)^(n + 1)* BernoulliB[2*n]> 0 |
Failure | Failure | Successful | Successful |
| 24.2.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{te^{xt}}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullipolyB{n}@{x}\frac{t^{n}}{n!}} | (t*exp(x*t))/(exp(t)- 1)= sum(bernoulli(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity) |
Divide[t*Exp[x*t],Exp[t]- 1]= Sum[BernoulliB[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 24.2.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \BernoullipolyB{n}@{0}} | bernoulli(n)= bernoulli(n, 0) |
BernoulliB[n]= BernoulliB[n, 0] |
Successful | Successful | - | - |
| 24.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2e^{t}}{e^{2t}+1} = \sum_{n=0}^{\infty}\EulernumberE{n}\frac{t^{n}}{n!}} | Error |
Divide[2*Exp[t],Exp[2*t]+ 1]= Sum[EulerE[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] |
Error | Successful | - | - |
| 24.2#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulernumberE{2n+1} = 0} | Error |
EulerE[2*n + 1]= 0 |
Error | Failure | - | Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 24.2#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\EulernumberE{2n} > 0} | Error |
(- 1)^(n)* EulerE[2*n]> 0 |
Error | Failure | - | Successful |
| 24.2.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2e^{xt}}{e^{t}+1} = \sum_{n=0}^{\infty}\EulerpolyE{n}@{x}\frac{t^{n}}{n!}} | (2*exp(x*t))/(exp(t)+ 1)= sum(euler(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity) |
Divide[2*Exp[x*t],Exp[t]+ 1]= Sum[EulerE[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] |
Failure | Successful | Skip | - |
| 24.2.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulernumberE{n} = 2^{n}\EulerpolyE{n}@{\tfrac{1}{2}}} | Error |
EulerE[n]= (2)^(n)* EulerE[n, Divide[1,2]] |
Error | Successful | - | - |
| 24.4.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{x+1}-\BernoullipolyB{n}@{x} = nx^{n-1}} | bernoulli(n, x + 1)- bernoulli(n, x)= n*(x)^(n - 1) |
BernoulliB[n, x + 1]- BernoulliB[n, x]= n*(x)^(n - 1) |
Successful | Successful | - | - |
| 24.4.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n}@{x+1}+\EulerpolyE{n}@{x} = 2x^{n}} | euler(n, x + 1)+ euler(n, x)= 2*(x)^(n) |
EulerE[n, x + 1]+ EulerE[n, x]= 2*(x)^(n) |
Successful | Successful | - | - |
| 24.4.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{1-x} = (-1)^{n}\BernoullipolyB{n}@{x}} | bernoulli(n, 1 - x)=(- 1)^(n)* bernoulli(n, x) |
BernoulliB[n, 1 - x]=(- 1)^(n)* BernoulliB[n, x] |
Successful | Failure | - | Successful |
| 24.4.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n}@{1-x} = (-1)^{n}\EulerpolyE{n}@{x}} | euler(n, 1 - x)=(- 1)^(n)* euler(n, x) |
EulerE[n, 1 - x]=(- 1)^(n)* EulerE[n, x] |
Successful | Failure | - | Successful |
| 24.4.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\BernoullipolyB{n}@{-x} = \BernoullipolyB{n}@{x}+nx^{n-1}} | (- 1)^(n)* bernoulli(n, - x)= bernoulli(n, x)+ n*(x)^(n - 1) |
(- 1)^(n)* BernoulliB[n, - x]= BernoulliB[n, x]+ n*(x)^(n - 1) |
Failure | Failure | Successful | Successful |
| 24.4.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n+1}\EulerpolyE{n}@{-x} = \EulerpolyE{n}@{x}-2x^{n}} | (- 1)^(n + 1)* euler(n, - x)= euler(n, x)- 2*(x)^(n) |
(- 1)^(n + 1)* EulerE[n, - x]= EulerE[n, x]- 2*(x)^(n) |
Failure | Failure | Successful | Successful |
| 24.4.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{m}k^{n} = \frac{\BernoullipolyB{n+1}@{m+1}-\BernoullinumberB{n+1}}{n+1}} | sum((k)^(n), k = 1..m)=(bernoulli(n + 1, m + 1)- bernoulli(n + 1))/(n + 1) |
Sum[(k)^(n), {k, 1, m}]=Divide[BernoulliB[n + 1, m + 1]- BernoulliB[n + 1],n + 1] |
Failure | Failure | Skip | Successful |
| 24.4.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{m}(-1)^{m-k}k^{n} = \frac{\EulerpolyE{n}@{m+1}+(-1)^{m}\EulerpolyE{n}@{0}}{2}} | sum((- 1)^(m - k)* (k)^(n), k = 1..m)=(euler(n, m + 1)+(- 1)^(m)* euler(n, 0))/(2) |
Sum[(- 1)^(m - k)* (k)^(n), {k, 1, m}]=Divide[EulerE[n, m + 1]+(- 1)^(m)* EulerE[n, 0],2] |
Failure | Failure | Skip | Successful |
| 24.4.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{m-1}(a+dk)^{n} = {\frac{d^{n}}{n+1}\left(\BernoullipolyB{n+1}@{m+\frac{a}{d}}-\BernoullipolyB{n+1}@{\frac{a}{d}}\right)}} | sum((a + d*k)^(n), k = 0..m - 1)=((d)^(n))/(n + 1)*(bernoulli(n + 1, m +(a)/(d))- bernoulli(n + 1, (a)/(d))) |
Sum[(a + d*k)^(n), {k, 0, m - 1}]=Divide[(d)^(n),n + 1]*(BernoulliB[n + 1, m +Divide[a,d]]- BernoulliB[n + 1, Divide[a,d]]) |
Failure | Failure | Skip | Skip |
| 24.4.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n} = {\frac{d^{n}}{2}\left((-1)^{m-1}\EulerpolyE{n}@{m+\frac{a}{d}}+\EulerpolyE{n}@{\frac{a}{d}}\right)}} | sum((- 1)^(k)*(a + d*k)^(n), k = 0..m - 1)=((d)^(n))/(2)*((- 1)^(m - 1)* euler(n, m +(a)/(d))+ euler(n, (a)/(d))) |
Sum[(- 1)^(k)*(a + d*k)^(n), {k, 0, m - 1}]=Divide[(d)^(n),2]*((- 1)^(m - 1)* EulerE[n, m +Divide[a,d]]+ EulerE[n, Divide[a,d]]) |
Failure | Failure | Skip | Skip |
| 24.4.E21 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{x} = 2^{n-1}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x}+\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}\right)} | bernoulli(n, x)= (2)^(n - 1)*(bernoulli(n, (1)/(2)*x)+ bernoulli(n, (1)/(2)*x +(1)/(2))) |
BernoulliB[n, x]= (2)^(n - 1)*(BernoulliB[n, Divide[1,2]*x]+ BernoulliB[n, Divide[1,2]*x +Divide[1,2]]) |
Failure | Failure | Successful | Successful |
| 24.4.E22 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n-1}@{x} = \frac{2}{n}\left(\BernoullipolyB{n}@{x}-2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)} | euler(n - 1, x)=(2)/(n)*(bernoulli(n, x)- (2)^(n)* bernoulli(n, (1)/(2)*x)) |
EulerE[n - 1, x]=Divide[2,n]*(BernoulliB[n, x]- (2)^(n)* BernoulliB[n, Divide[1,2]*x]) |
Failure | Failure | Successful | Successful |
| 24.4.E23 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n-1}@{x} = \frac{2^{n}}{n}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}-\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)} | euler(n - 1, x)=((2)^(n))/(n)*(bernoulli(n, (1)/(2)*x +(1)/(2))- bernoulli(n, (1)/(2)*x)) |
EulerE[n - 1, x]=Divide[(2)^(n),n]*(BernoulliB[n, Divide[1,2]*x +Divide[1,2]]- BernoulliB[n, Divide[1,2]*x]) |
Failure | Failure | Successful | Successful |
| 24.4.E25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{0} = (-1)^{n}\BernoullipolyB{n}@{1}} | bernoulli(n, 0)=(- 1)^(n)* bernoulli(n, 1) |
BernoulliB[n, 0]=(- 1)^(n)* BernoulliB[n, 1] |
Failure | Failure | Successful | Successful |
| 24.4.E25 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\BernoullipolyB{n}@{1} = \BernoullinumberB{n}} | (- 1)^(n)* bernoulli(n, 1)= bernoulli(n) |
(- 1)^(n)* BernoulliB[n, 1]= BernoulliB[n] |
Failure | Failure | Successful | Successful |
| 24.4.E26 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n}@{0} = -\EulerpolyE{n}@{1}} | euler(n, 0)= - euler(n, 1) |
EulerE[n, 0]= - EulerE[n, 1] |
Successful | Successful | - | - |
| 24.4.E26 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\EulerpolyE{n}@{1} = -\frac{2}{n+1}(2^{n+1}-1)\BernoullinumberB{n+1}} | - euler(n, 1)= -(2)/(n + 1)*((2)^(n + 1)- 1)* bernoulli(n + 1) |
- EulerE[n, 1]= -Divide[2,n + 1]*((2)^(n + 1)- 1)* BernoulliB[n + 1] |
Failure | Failure | Error | Successful |
| 24.4.E27 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{\tfrac{1}{2}} = -(1-2^{1-n})\BernoullinumberB{n}} | bernoulli(n, (1)/(2))= -(1 - (2)^(1 - n))* bernoulli(n) |
BernoulliB[n, Divide[1,2]]= -(1 - (2)^(1 - n))* BernoulliB[n] |
Successful | Successful | - | - |
| 24.4.E28 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n}@{\tfrac{1}{2}} = 2^{-n}\EulernumberE{n}} | Error |
EulerE[n, Divide[1,2]]= (2)^(- n)* EulerE[n] |
Error | Successful | - | - |
| 24.4.E29 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n}@{\tfrac{1}{3}} = \BernoullipolyB{2n}@{\tfrac{2}{3}}} | bernoulli(2*n, (1)/(3))= bernoulli(2*n, (2)/(3)) |
BernoulliB[2*n, Divide[1,3]]= BernoulliB[2*n, Divide[2,3]] |
Failure | Failure | Successful | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.4.E29 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n}@{\tfrac{2}{3}} = -\tfrac{1}{2}(1-3^{1-2n})\BernoullinumberB{2n}} | bernoulli(2*n, (2)/(3))= -(1)/(2)*(1 - (3)^(1 - 2*n))* bernoulli(2*n) |
BernoulliB[2*n, Divide[2,3]]= -Divide[1,2]*(1 - (3)^(1 - 2*n))* BernoulliB[2*n] |
Failure | Failure | Successful | Successful |
| 24.4.E30 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n-1}@{\tfrac{1}{3}} = -\EulerpolyE{2n-1}@{\tfrac{2}{3}}} | euler(2*n - 1, (1)/(3))= - euler(2*n - 1, (2)/(3)) |
EulerE[2*n - 1, Divide[1,3]]= - EulerE[2*n - 1, Divide[2,3]] |
Failure | Failure | Successful | Fail
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.0, 2.8284271247461903] <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.4.E30 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\EulerpolyE{2n-1}@{\tfrac{2}{3}} = -\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}\BernoullinumberB{2n}} | - euler(2*n - 1, (2)/(3))= -((1 - (3)^(1 - 2*n))*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n) |
- EulerE[2*n - 1, Divide[2,3]]= -Divide[(1 - (3)^(1 - 2*n))*((2)^(2*n)- 1),2*n]*BernoulliB[2*n] |
Failure | Failure | Successful | Successful |
| 24.4.E31 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{\tfrac{1}{4}} = (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}}} | bernoulli(n, (1)/(4))=(- 1)^(n)* bernoulli(n, (3)/(4)) |
BernoulliB[n, Divide[1,4]]=(- 1)^(n)* BernoulliB[n, Divide[3,4]] |
Failure | Successful | Successful | - |
| 24.4.E31 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}} = -\frac{1-2^{1-n}}{2^{n}}\BernoullinumberB{n}-\frac{n}{4^{n}}\EulernumberE{n-1}} | Error |
(- 1)^(n)* BernoulliB[n, Divide[3,4]]= -Divide[1 - (2)^(1 - n),(2)^(n)]*BernoulliB[n]-Divide[n,(4)^(n)]*EulerE[n - 1] |
Error | Failure | - | Successful |
| 24.4.E32 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n}@{\tfrac{1}{6}} = \BernoullipolyB{2n}@{\tfrac{5}{6}}} | bernoulli(2*n, (1)/(6))= bernoulli(2*n, (5)/(6)) |
BernoulliB[2*n, Divide[1,6]]= BernoulliB[2*n, Divide[5,6]] |
Failure | Failure | Successful | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.4.E32 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n}@{\tfrac{5}{6}} = \tfrac{1}{2}(1-2^{1-2n})(1-3^{1-2n})\BernoullinumberB{2n}} | bernoulli(2*n, (5)/(6))=(1)/(2)*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))* bernoulli(2*n) |
BernoulliB[2*n, Divide[5,6]]=Divide[1,2]*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))* BernoulliB[2*n] |
Failure | Failure | Successful | Successful |
| 24.4.E33 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n}@{\tfrac{1}{6}} = \EulerpolyE{2n}@{\tfrac{5}{6}}} | euler(2*n, (1)/(6))= euler(2*n, (5)/(6)) |
EulerE[2*n, Divide[1,6]]= EulerE[2*n, Divide[5,6]] |
Failure | Failure | Successful | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.4.E33 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n}@{\tfrac{5}{6}} = \frac{1+3^{-2n}}{2^{2n+1}}\EulernumberE{2n}} | Error |
EulerE[2*n, Divide[5,6]]=Divide[1 + (3)^(- 2*n),(2)^(2*n + 1)]*EulerE[2*n] |
Error | Failure | - | Successful |
| 24.4.E34 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\BernoullipolyB{n}@{x} = n\BernoullipolyB{n-1}@{x}} | diff(bernoulli(n, x), x)= n*bernoulli(n - 1, x) |
D[BernoulliB[n, x], x]= n*BernoulliB[n - 1, x] |
Successful | Successful | - | - |
| 24.4.E35 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{x}\EulerpolyE{n}@{x} = n\EulerpolyE{n-1}@{x}} | diff(euler(n, x), x)= n*euler(n - 1, x) |
D[EulerE[n, x], x]= n*EulerE[n - 1, x] |
Successful | Successful | - | - |
| 24.4.E37 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{x+h} = (B(x)+h)^{n}} | bernoulli(n, x + h)=(B*(x)+ h)^(n) |
BernoulliB[n, x + h]=(B*(x)+ h)^(n) |
Failure | Failure | Fail -.9142135620-1.414213562*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 1}-1.328427124-2.828427124*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 2}-1.742640686-4.242640686*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 3}1.580880229-10.58578643*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Fail
Complex[-0.9142135623730951, -1.4142135623730951] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}Complex[-1.3284271247461903, -2.8284271247461903] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}Complex[-1.7426406871192857, -4.242640687119286] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}Complex[1.580880229039761, -10.585786437626906] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}... skip entries to safe data |
| 24.4.E39 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n}@{x+h} = (E(x)+h)^{n}} | euler(n, x + h)=(E*(x)+ h)^(n) |
EulerE[n, x + h]=(E*(x)+ h)^(n) |
Failure | Failure | Fail -.9142135620-1.414213562*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 1}-1.328427124-2.828427124*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 2}-1.742640686-4.242640686*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 3}1.414213562-10.58578643*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Fail
-2.218281828459045 <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}-3.93656365691809 <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}-5.654845485377136 <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}Complex[-13.66330459287579, -6.27424849394514] <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}... skip entries to safe data |
| 24.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n}\frac{2^{2k}\BernoullinumberB{2k}}{(2k)!(2n+1-2k)!} = \frac{1}{(2n)!}} | sum(((2)^(2*k)* bernoulli(2*k))/(factorial(2*k)*factorial(2*n + 1 - 2*k)), k = 0..n)=(1)/(factorial(2*n)) |
Sum[Divide[(2)^(2*k)* BernoulliB[2*k],(2*k)!*(2*n + 1 - 2*k)!], {k, 0, n}]=Divide[1,(2*n)!] |
Failure | Failure | Skip | Successful |
| 24.6.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\binom{n+1}{k-j}}\Bigg{/}{\binom{n}{k}}} | bernoulli(n)=(1)/(n + 1)*sum(sum((- 1)^(j)* (j)^(n)*binomial(n + 1,k - j)/binomial(n,k), j = 1..k), k = 1..n) |
BernoulliB[n]=Divide[1,n + 1]*Sum[Sum[(- 1)^(j)* (j)^(n)*Binomial[n + 1,k - j]/Binomial[n,k], {j, 1, k}], {k, 1, n}] |
Failure | Failure | Skip | Successful |
| 24.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = (-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}+1}\diff{t}} | bernoulli(2*n)=(- 1)^(n + 1)*(4*n)/(1 - (2)^(1 - 2*n))*int(((t)^(2*n - 1))/(exp(2*Pi*t)+ 1), t = 0..infinity) |
BernoulliB[2*n]=(- 1)^(n + 1)*Divide[4*n,1 - (2)^(1 - 2*n)]*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]+ 1], {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 24.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}+1}\diff{t} = (-1)^{n+1}\frac{2n}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\sech@{\pi t}\diff{t}} | (- 1)^(n + 1)*(4*n)/(1 - (2)^(1 - 2*n))*int(((t)^(2*n - 1))/(exp(2*Pi*t)+ 1), t = 0..infinity)=(- 1)^(n + 1)*(2*n)/(1 - (2)^(1 - 2*n))*int((t)^(2*n - 1)* exp(- Pi*t)*sech(Pi*t), t = 0..infinity) |
(- 1)^(n + 1)*Divide[4*n,1 - (2)^(1 - 2*n)]*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]+ 1], {t, 0, Infinity}]=(- 1)^(n + 1)*Divide[2*n,1 - (2)^(1 - 2*n)]*Integrate[(t)^(2*n - 1)* Exp[- Pi*t]*Sech[Pi*t], {t, 0, Infinity}] |
Successful | Failure | - | Skip |
| 24.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = (-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}-1}\diff{t}} | bernoulli(2*n)=(- 1)^(n + 1)* 4*n*int(((t)^(2*n - 1))/(exp(2*Pi*t)- 1), t = 0..infinity) |
BernoulliB[2*n]=(- 1)^(n + 1)* 4*n*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]- 1], {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 24.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}-1}\diff{t} = (-1)^{n+1}2n\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\csch@{\pi t}\diff{t}} | (- 1)^(n + 1)* 4*n*int(((t)^(2*n - 1))/(exp(2*Pi*t)- 1), t = 0..infinity)=(- 1)^(n + 1)* 2*n*int((t)^(2*n - 1)* exp(- Pi*t)*csch(Pi*t), t = 0..infinity) |
(- 1)^(n + 1)* 4*n*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]- 1], {t, 0, Infinity}]=(- 1)^(n + 1)* 2*n*Integrate[(t)^(2*n - 1)* Exp[- Pi*t]*Csch[Pi*t], {t, 0, Infinity}] |
Successful | Failure | - | Skip |
| 24.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = (-1)^{n+1}\frac{\pi}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n}\sech^{2}@{\pi t}\diff{t}} | bernoulli(2*n)=(- 1)^(n + 1)*(Pi)/(1 - (2)^(1 - 2*n))*int((t)^(2*n)* (sech(Pi*t))^(2), t = 0..infinity) |
BernoulliB[2*n]=(- 1)^(n + 1)*Divide[Pi,1 - (2)^(1 - 2*n)]*Integrate[(t)^(2*n)* (Sech[Pi*t])^(2), {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 24.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = (-1)^{n+1}\pi\int_{0}^{\infty}t^{2n}\csch^{2}@{\pi t}\diff{t}} | bernoulli(2*n)=(- 1)^(n + 1)* Pi*int((t)^(2*n)* (csch(Pi*t))^(2), t = 0..infinity) |
BernoulliB[2*n]=(- 1)^(n + 1)* Pi*Integrate[(t)^(2*n)* (Csch[Pi*t])^(2), {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 24.7.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = (-1)^{n}\frac{2n(2n-1)}{\pi}\*\int_{0}^{\infty}t^{2n-2}\ln@{1-e^{-2\pi t}}\diff{t}} | bernoulli(2*n)=(- 1)^(n)*(2*n*(2*n - 1))/(Pi)* int((t)^(2*n - 2)* ln(1 - exp(- 2*Pi*t)), t = 0..infinity) |
BernoulliB[2*n]=(- 1)^(n)*Divide[2*n*(2*n - 1),Pi]* Integrate[(t)^(2*n - 2)* Log[1 - Exp[- 2*Pi*t]], {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 24.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulernumberE{2n} = (-1)^{n}2^{2n+1}\int_{0}^{\infty}t^{2n}\sech@{\pi t}\diff{t}} | Error |
EulerE[2*n]=(- 1)^(n)* (2)^(2*n + 1)* Integrate[(t)^(2*n)* Sech[Pi*t], {t, 0, Infinity}] |
Error | Failure | - | Skip |
| 24.7.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n}@{x} = (-1)^{n+1}2n\*\int_{0}^{\infty}\frac{\cos@{2\pi x}-e^{-2\pi t}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n-1}\diff{t}} | bernoulli(2*n, x)=(- 1)^(n + 1)* 2*n * int((cos(2*Pi*x)- exp(- 2*Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n - 1), t = 0..infinity) |
BernoulliB[2*n, x]=(- 1)^(n + 1)* 2*n * Integrate[Divide[Cos[2*Pi*x]- Exp[- 2*Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n - 1), {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 24.7.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n+1}@{x} = (-1)^{n+1}(2n+1)\*\int_{0}^{\infty}\frac{\sin@{2\pi x}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n}\diff{t}} | bernoulli(2*n + 1, x)=(- 1)^(n + 1)*(2*n + 1)* int((sin(2*Pi*x))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n), t = 0..infinity) |
BernoulliB[2*n + 1, x]=(- 1)^(n + 1)*(2*n + 1)* Integrate[Divide[Sin[2*Pi*x],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n), {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 24.7.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n}@{x} = (-1)^{n}4\int_{0}^{\infty}\frac{\sin@{\pi x}\cosh@{\pi t}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n}\diff{t}} | euler(2*n, x)=(- 1)^(n)* 4*int((sin(Pi*x)*cosh(Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n), t = 0..infinity) |
EulerE[2*n, x]=(- 1)^(n)* 4*Integrate[Divide[Sin[Pi*x]*Cosh[Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n), {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 24.7.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n+1}@{x} = (-1)^{n+1}4\*\int_{0}^{\infty}\frac{\cos@{\pi x}\sinh@{\pi t}}{\cosh@{2\pi t}-\cos@{2\pi x}}t^{2n+1}\diff{t}} | euler(2*n + 1, x)=(- 1)^(n + 1)* 4 * int((cos(Pi*x)*sinh(Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n + 1), t = 0..infinity) |
EulerE[2*n + 1, x]=(- 1)^(n + 1)* 4 * Integrate[Divide[Cos[Pi*x]*Sinh[Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n + 1), {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 24.7.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{x} = \frac{1}{2\pi i}\int_{-c-i\infty}^{-c+i\infty}(x+t)^{n}\left(\frac{\pi}{\sin@{\pi t}}\right)^{2}\diff{t}} | bernoulli(n, x)=(1)/(2*Pi*I)*int((x + t)^(n)*((Pi)/(sin(Pi*t)))^(2), t = - c - I*infinity..- c + I*infinity) |
BernoulliB[n, x]=Divide[1,2*Pi*I]*Integrate[(x + t)^(n)*(Divide[Pi,Sin[Pi*t]])^(2), {t, - c - I*Infinity, - c + I*Infinity}] |
Failure | Failure | Skip | Fail
Complex[1.1891344833338187, 1.6392926414123716] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.6392926414123716, 1.6392926414123716] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[1.6392926414123716, 1.1891344833338187] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.1891344833338187, 1.1891344833338187] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.8.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n}@{x} = (-1)^{n+1}\frac{2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos@{2\pi kx}}{k^{2n}}} | bernoulli(2*n, x)=(- 1)^(n + 1)*(2*factorial(2*n))/((2*Pi)^(2*n))*sum((cos(2*Pi*k*x))/((k)^(2*n)), k = 1..infinity) |
BernoulliB[2*n, x]=(- 1)^(n + 1)*Divide[2*(2*n)!,(2*Pi)^(2*n)]*Sum[Divide[Cos[2*Pi*k*x],(k)^(2*n)], {k, 1, Infinity}] |
Failure | Failure | Skip | Fail
2.0 <- {Rule[n, 1], Rule[x, 2]}6.0 <- {Rule[n, 1], Rule[x, 3]}4.0 <- {Rule[n, 2], Rule[x, 2]}36.0 <- {Rule[n, 2], Rule[x, 3]}... skip entries to safe data |
| 24.8.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n+1}@{x} = (-1)^{n+1}\frac{2(2n+1)!}{(2\pi)^{2n+1}}\sum_{k=1}^{\infty}\frac{\sin@{2\pi kx}}{k^{2n+1}}} | bernoulli(2*n + 1, x)=(- 1)^(n + 1)*(2*factorial(2*n + 1))/((2*Pi)^(2*n + 1))*sum((sin(2*Pi*k*x))/((k)^(2*n + 1)), k = 1..infinity) |
BernoulliB[2*n + 1, x]=(- 1)^(n + 1)*Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)]*Sum[Divide[Sin[2*Pi*k*x],(k)^(2*n + 1)], {k, 1, Infinity}] |
Failure | Failure | Skip | Fail
3.0 <- {Rule[n, 1], Rule[x, 2]}15.0 <- {Rule[n, 1], Rule[x, 3]}5.0 <- {Rule[n, 2], Rule[x, 2]}85.0 <- {Rule[n, 2], Rule[x, 3]}... skip entries to safe data |
| 24.8.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n}@{x} = (-1)^{n}\frac{4(2n)!}{\pi^{2n+1}}\sum_{k=0}^{\infty}\frac{\sin@{(2k+1)\pi x}}{(2k+1)^{2n+1}}} | euler(2*n, x)=(- 1)^(n)*(4*factorial(2*n))/((Pi)^(2*n + 1))*sum((sin((2*k + 1)* Pi*x))/((2*k + 1)^(2*n + 1)), k = 0..infinity) |
EulerE[2*n, x]=(- 1)^(n)*Divide[4*(2*n)!,(Pi)^(2*n + 1)]*Sum[Divide[Sin[(2*k + 1)* Pi*x],(2*k + 1)^(2*n + 1)], {k, 0, Infinity}] |
Failure | Failure | Skip | Fail
2.0 <- {Rule[n, 1], Rule[x, 2]}6.0 <- {Rule[n, 1], Rule[x, 3]}2.0 <- {Rule[n, 2], Rule[x, 2]}30.0 <- {Rule[n, 2], Rule[x, 3]}... skip entries to safe data |
| 24.8.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n-1}@{x} = (-1)^{n}\frac{4(2n-1)!}{\pi^{2n}}\sum_{k=0}^{\infty}\frac{\cos@{(2k+1)\pi x}}{(2k+1)^{2n}}} | euler(2*n - 1, x)=(- 1)^(n)*(4*factorial(2*n - 1))/((Pi)^(2*n))*sum((cos((2*k + 1)* Pi*x))/((2*k + 1)^(2*n)), k = 0..infinity) |
EulerE[2*n - 1, x]=(- 1)^(n)*Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Sum[Divide[Cos[(2*k + 1)* Pi*x],(2*k + 1)^(2*n)], {k, 0, Infinity}] |
Failure | Failure | Skip | Fail
2.0 <- {Rule[n, 1], Rule[x, 2]}2.0 <- {Rule[n, 1], Rule[x, 3]}2.0 <- {Rule[n, 2], Rule[x, 2]}14.0 <- {Rule[n, 2], Rule[x, 3]}... skip entries to safe data |
| 24.8.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{4n+2} = (8n+4)\sum_{k=1}^{\infty}\frac{k^{4n+1}}{e^{2\pi k}-1}} | bernoulli(4*n + 2)=(8*n + 4)* sum(((k)^(4*n + 1))/(exp(2*Pi*k)- 1), k = 1..infinity) |
BernoulliB[4*n + 2]=(8*n + 4)* Sum[Divide[(k)^(4*n + 1),Exp[2*Pi*k]- 1], {k, 1, Infinity}] |
Failure | Failure | Skip | Skip |
| 24.8.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = \frac{(-1)^{n+1}4n}{2^{2n}-1}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{\pi k}+(-1)^{k+n}}} | bernoulli(2*n)=((- 1)^(n + 1)* 4*n)/((2)^(2*n)- 1)*sum(((k)^(2*n - 1))/(exp(Pi*k)+(- 1)^(k + n)), k = 1..infinity) |
BernoulliB[2*n]=Divide[(- 1)^(n + 1)* 4*n,(2)^(2*n)- 1]*Sum[Divide[(k)^(2*n - 1),Exp[Pi*k]+(- 1)^(k + n)], {k, 1, Infinity}] |
Failure | Failure | Skip | Skip |
| 24.8.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\BernoullinumberB{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}\right) = \alpha^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1}} | (bernoulli(2*n))/(4*n)*((alpha)^(n)-(- beta)^(n))= (alpha)^(n)* sum(((k)^(2*n - 1))/(exp(2*alpha*k)- 1), k = 1..infinity)-(- beta)^(n)* sum(((k)^(2*n - 1))/(exp(2*beta*k)- 1), k = 1..infinity) |
Divide[BernoulliB[2*n],4*n]*((\[Alpha])^(n)-(- \[Beta])^(n))= (\[Alpha])^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Alpha]*k]- 1], {k, 1, Infinity}]-(- \[Beta])^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Beta]*k]- 1], {k, 1, Infinity}] |
Failure | Failure | Skip | Error |
| 24.8.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulernumberE{2n} = (-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh@{\tfrac{1}{2}\pi k}}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1}} | Error |
EulerE[2*n]=(- 1)^(n)* Sum[Divide[(k)^(2*n),Cosh[Divide[1,2]*Pi*k]], {k, 1, Infinity}]- 4*Sum[Divide[(- 1)^(k)*(2*k + 1)^(2*n),Exp[2*Pi*(2*k + 1)]- 1], {k, 0, Infinity}] |
Error | Failure | - | Skip |
| 24.9.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\BernoullinumberB{2n}| > |\BernoullipolyB{2n}@{x}|} | abs(bernoulli(2*n))>abs(bernoulli(2*n, x)) |
Abs[BernoulliB[2*n]]>Abs[BernoulliB[2*n, x]] |
Failure | Failure | Skip | Successful |
| 24.9.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2-2^{1-2n})|\BernoullinumberB{2n}| >= |\BernoullipolyB{2n}@{x}-\BernoullinumberB{2n}|} | (2 - (2)^(1 - 2*n))*abs(bernoulli(2*n))> =abs(bernoulli(2*n, x)- bernoulli(2*n)) |
(2 - (2)^(1 - 2*n))*Abs[BernoulliB[2*n]]> =Abs[BernoulliB[2*n, x]- BernoulliB[2*n]] |
Failure | Failure | Skip | Successful |
| 24.9.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 4^{-n}|\EulernumberE{2n}| > (-1)^{n}\EulerpolyE{2n}@{x}} | Error |
(4)^(- n)*Abs[EulerE[2*n]]>(- 1)^(n)* EulerE[2*n, x] |
Error | Failure | - | Successful |
| 24.9.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\EulerpolyE{2n}@{x} > 0} | (- 1)^(n)* euler(2*n, x)> 0 |
(- 1)^(n)* EulerE[2*n, x]> 0 |
Failure | Failure | Fail 0. < 0. <- {n = 1, x = 1}0. < -2. <- {n = 1, x = 2}0. < -6. <- {n = 1, x = 3}0. < 0. <- {n = 2, x = 1}... skip entries to safe data |
Successful |
| 24.9.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2(2n+1)!}{(2\pi)^{2n+1}} > (-1)^{n+1}\BernoullipolyB{2n+1}@{x}} | (2*factorial(2*n + 1))/((2*Pi)^(2*n + 1))>(- 1)^(n + 1)* bernoulli(2*n + 1, x) |
Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)]>(- 1)^(n + 1)* BernoulliB[2*n + 1, x] |
Failure | Failure | Fail 3. < .4837730163e-1 <- {n = 1, x = 2}15. < .4837730163e-1 <- {n = 1, x = 3}7. < .2607362729e-1 <- {n = 3, x = 2}455. < .2607362729e-1 <- {n = 3, x = 3} |
Successful |
| 24.9.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n+1}\BernoullipolyB{2n+1}@{x} > 0} | (- 1)^(n + 1)* bernoulli(2*n + 1, x)> 0 |
(- 1)^(n + 1)* BernoulliB[2*n + 1, x]> 0 |
Failure | Failure | Fail 0. < 0. <- {n = 1, x = 1}0. < 0. <- {n = 2, x = 1}0. < -5. <- {n = 2, x = 2}0. < -85. <- {n = 2, x = 3}... skip entries to safe data |
Successful |
| 24.9.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2} > (-1)^{n}\EulerpolyE{2n-1}@{x}} | (4*factorial(2*n - 1))/((Pi)^(2*n))*((2)^(2*n)- 1)/((2)^(2*n)- 2)>(- 1)^(n)* euler(2*n - 1, x) |
Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Divide[(2)^(2*n)- 1,(2)^(2*n)- 2]>(- 1)^(n)* EulerE[2*n - 1, x] |
Failure | Failure | Fail 2.250000000 < .2639824007 <- {n = 2, x = 2}13.75000000 < .2639824007 <- {n = 2, x = 3} |
Successful |
| 24.9.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\EulerpolyE{2n-1}@{x} > 0} | (- 1)^(n)* euler(2*n - 1, x)> 0 |
(- 1)^(n)* EulerE[2*n - 1, x]> 0 |
Failure | Failure | Fail 0. < -.5000000000 <- {n = 1, x = 1}0. < -1.500000000 <- {n = 1, x = 2}0. < -2.500000000 <- {n = 1, x = 3}0. < -.2500000000 <- {n = 2, x = 1}... skip entries to safe data |
Successful |
| 24.9.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n} > (-1)^{n+1}\BernoullinumberB{2n}} | 5*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n)>(- 1)^(n + 1)* bernoulli(2*n) |
5*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n)>(- 1)^(n + 1)* BernoulliB[2*n] |
Failure | Failure | Fail .1666666667 < .1215223702 <- {n = 1} |
Successful |
| 24.9.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n+1}\BernoullinumberB{2n} > 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}} | (- 1)^(n + 1)* bernoulli(2*n)> 4*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n) |
(- 1)^(n + 1)* BernoulliB[2*n]> 4*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n) |
Failure | Failure | Successful | Successful |
| 24.9.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}\right) > (-1)^{n}\EulernumberE{2n}} | Error |
8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)*(1 +Divide[1,12*n])>(- 1)^(n)* EulerE[2*n] |
Error | Failure | - | Successful |
| 24.9.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\EulernumberE{2n} > 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}} | Error |
(- 1)^(n)* EulerE[2*n]> 8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n) |
Error | Failure | - | Successful |
| 24.9.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}} >= (-1)^{n+1}\BernoullinumberB{2n}\geq\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}} | (2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(beta - 2*n))> =(- 1)^(n + 1)* bernoulli(2*n)>=(2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(- 2*n)) |
Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(\[Beta]- 2*n)]> =(- 1)^(n + 1)* BernoulliB[2*n]>=Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(- 2*n)] |
Failure | Failure | Error | Successful |
| 24.9.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \beta = 2+\frac{\ln@{1-6\pi^{-2}}}{\ln@@{2}}} | beta = 2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2)) |
\[Beta]= 2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]] |
Failure | Failure | Fail .765019736+1.414213562*I <- {beta = 2^(1/2)+I*2^(1/2)}.765019736-1.414213562*I <- {beta = 2^(1/2)-I*2^(1/2)}-2.063407388-1.414213562*I <- {beta = -2^(1/2)-I*2^(1/2)}-2.063407388+1.414213562*I <- {beta = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[0.7650197375731231, 1.4142135623730951] <- {Rule[β, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.7650197375731231, -1.4142135623730951] <- {Rule[β, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.0634073871730667, -1.4142135623730951] <- {Rule[β, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.0634073871730667, 1.4142135623730951] <- {Rule[β, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 24.9.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{4^{n+1}(2n)!}{\pi^{2n+1}} > (-1)^{n}\EulernumberE{2n}} | Error |
Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)]>(- 1)^(n)* EulerE[2*n] |
Error | Failure | - | Successful |
| 24.9.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\EulernumberE{2n} > \frac{4^{n+1}(2n)!}{\pi^{2n+1}}\frac{1}{1+3^{-1-2n}}} | Error |
(- 1)^(n)* EulerE[2*n]>Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)]*Divide[1,1 + (3)^(- 1 - 2*n)] |
Error | Failure | - | Successful |
| 24.13.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{x}^{x+1}\BernoullipolyB{n}@{t}\diff{t} = x^{n}} | int(bernoulli(n, t), t = x..x + 1)= (x)^(n) |
Integrate[BernoulliB[n, t], {t, x, x + 1}]= (x)^(n) |
Failure | Failure | Skip | Successful |
| 24.13.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{x}^{x+(1/2)}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulerpolyE{n}@{2x}}{2^{n+1}}} | int(bernoulli(n, t), t = x..x +(1/ 2))=(euler(n, 2*x))/((2)^(n + 1)) |
Integrate[BernoulliB[n, t], {t, x, x +(1/ 2)}]=Divide[EulerE[n, 2*x],(2)^(n + 1)] |
Failure | Failure | Skip | Successful |
| 24.13.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1/2}\BernoullipolyB{n}@{t}\diff{t} = \frac{1-2^{n+1}}{2^{n}}\frac{\BernoullinumberB{n+1}}{n+1}} | int(bernoulli(n, t), t = 0..1/ 2)=(1 - (2)^(n + 1))/((2)^(n))*(bernoulli(n + 1))/(n + 1) |
Integrate[BernoulliB[n, t], {t, 0, 1/ 2}]=Divide[1 - (2)^(n + 1),(2)^(n)]*Divide[BernoulliB[n + 1],n + 1] |
Failure | Failure | Skip | Successful |
| 24.13.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{1/4}^{3/4}\BernoullipolyB{n}@{t}\diff{t} = \frac{\EulernumberE{n}}{2^{2n+1}}} | Error |
Integrate[BernoulliB[n, t], {t, 1/ 4, 3/ 4}]=Divide[EulerE[n],(2)^(2*n + 1)] |
Error | Failure | - | Successful |
| 24.13.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\BernoullipolyB{n}@{t}\BernoullipolyB{m}@{t}\diff{t} = \frac{(-1)^{n-1}m!n!}{(m+n)!}\BernoullinumberB{m+n}} | int(bernoulli(n, t)*bernoulli(m, t), t = 0..1)=((- 1)^(n - 1)* factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n) |
Integrate[BernoulliB[n, t]*BernoulliB[m, t], {t, 0, 1}]=Divide[(- 1)^(n - 1)* (m)!*(n)!,(m + n)!]*BernoulliB[m + n] |
Failure | Failure | Skip | Successful |
| 24.13.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\EulerpolyE{n}@{t}\diff{t} = -2\frac{\EulerpolyE{n+1}@{0}}{n+1}} | int(euler(n, t), t = 0..1)= - 2*(euler(n + 1, 0))/(n + 1) |
Integrate[EulerE[n, t], {t, 0, 1}]= - 2*Divide[EulerE[n + 1, 0],n + 1] |
Failure | Failure | Skip | Successful |
| 24.13.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -2\frac{\EulerpolyE{n+1}@{0}}{n+1} = \frac{4(2^{n+2}-1)}{(n+1)(n+2)}\BernoullinumberB{n+2}} | - 2*(euler(n + 1, 0))/(n + 1)=(4*((2)^(n + 2)- 1))/((n + 1)*(n + 2))*bernoulli(n + 2) |
- 2*Divide[EulerE[n + 1, 0],n + 1]=Divide[4*((2)^(n + 2)- 1),(n + 1)*(n + 2)]*BernoulliB[n + 2] |
Failure | Failure | Successful | Successful |
| 24.13.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1/2}\EulerpolyE{2n}@{t}\diff{t} = -\frac{\EulerpolyE{2n+1}@{0}}{2n+1}} | int(euler(2*n, t), t = 0..1/ 2)= -(euler(2*n + 1, 0))/(2*n + 1) |
Integrate[EulerE[2*n, t], {t, 0, 1/ 2}]= -Divide[EulerE[2*n + 1, 0],2*n + 1] |
Failure | Failure | Skip | Successful |
| 24.13.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\frac{\EulerpolyE{2n+1}@{0}}{2n+1} = \frac{2(2^{2n+2}-1)\BernoullinumberB{2n+2}}{(2n+1)(2n+2)}} | -(euler(2*n + 1, 0))/(2*n + 1)=(2*((2)^(2*n + 2)- 1)* bernoulli(2*n + 2))/((2*n + 1)*(2*n + 2)) |
-Divide[EulerE[2*n + 1, 0],2*n + 1]=Divide[2*((2)^(2*n + 2)- 1)* BernoulliB[2*n + 2],(2*n + 1)*(2*n + 2)] |
Failure | Failure | Successful | Successful |
| 24.13.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1/2}\EulerpolyE{2n-1}@{t}\diff{t} = \frac{\EulernumberE{2n}}{n2^{2n+1}}} | Error |
Integrate[EulerE[2*n - 1, t], {t, 0, 1/ 2}]=Divide[EulerE[2*n],n*(2)^(2*n + 1)] |
Error | Failure | - | Successful |
| 24.13.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\EulerpolyE{n}@{t}\EulerpolyE{m}@{t}\diff{t} = (-1)^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}\BernoullinumberB{m+n+2}} | int(euler(n, t)*euler(m, t), t = 0..1)=(- 1)^(n)* 4*(((2)^(m + n + 2)- 1)* factorial(m)*factorial(n))/(factorial(m + n + 2))*bernoulli(m + n + 2) |
Integrate[EulerE[n, t]*EulerE[m, t], {t, 0, 1}]=(- 1)^(n)* 4*Divide[((2)^(m + n + 2)- 1)* (m)!*(n)!,(m + n + 2)!]*BernoulliB[m + n + 2] |
Failure | Failure | Skip | Successful |
| 24.14.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -2^{n+1}\EulerpolyE{n+1}@{0} = -2^{n+2}(1-2^{n+2})\frac{\BernoullinumberB{n+2}}{n+2}} | - (2)^(n + 1)* euler(n + 1, 0)= - (2)^(n + 2)*(1 - (2)^(n + 2))*(bernoulli(n + 2))/(n + 2) |
- (2)^(n + 1)* EulerE[n + 1, 0]= - (2)^(n + 2)*(1 - (2)^(n + 2))*Divide[BernoulliB[n + 2],n + 2] |
Failure | Failure | Successful | Successful |
| 24.14.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{j=0}^{m}\sum_{k=0}^{n}\binom{m}{j}\binom{n}{k}\frac{\BernoullinumberB{j}\BernoullinumberB{k}}{m+n-j-k+1} = (-1)^{m-1}\frac{m!n!}{(m+n)!}\BernoullinumberB{m+n}} | sum(sum(binomial(m,j)*binomial(n,k)*(bernoulli(j)*bernoulli(k))/(m + n - j - k + 1), k = 0..n), j = 0..m)=(- 1)^(m - 1)*(factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n) |
Sum[Sum[Binomial[m,j]*Binomial[n,k]*Divide[BernoulliB[j]*BernoulliB[k],m + n - j - k + 1], {k, 0, n}], {j, 0, m}]=(- 1)^(m - 1)*Divide[(m)!*(n)!,(m + n)!]*BernoulliB[m + n] |
Failure | Failure | Skip | Successful |
| 24.15.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{n} = 2(1-2^{n})\BernoullinumberB{n}} | G[n]= 2*(1 - (2)^(n))* bernoulli(n) |
Subscript[G, n]= 2*(1 - (2)^(n))* BernoulliB[n] |
Failure | Failure | Fail .414213562+1.414213562*I <- {G[n] = 2^(1/2)+I*2^(1/2), n = 1}2.414213562+1.414213562*I <- {G[n] = 2^(1/2)+I*2^(1/2), n = 2}1.414213562+1.414213562*I <- {G[n] = 2^(1/2)+I*2^(1/2), n = 3}.414213562-1.414213562*I <- {G[n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Successful |
| 24.15.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \tan@@{t} = \sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}} | tan(t)= sum(T[n]*((t)^(n))/(factorial(n)), n = 0..infinity) |
Tan[t]= Sum[Subscript[T, n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 24.15.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle T_{2n-1} = (-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}\BernoullinumberB{2n}} | T[2*n - 1]=(- 1)^(n - 1)*((2)^(2*n)*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n) |
Subscript[T, 2*n - 1]=(- 1)^(n - 1)*Divide[(2)^(2*n)*((2)^(2*n)- 1),2*n]*BernoulliB[2*n] |
Failure | Failure | Fail .414213562+1.414213562*I <- {T[2*n-1] = 2^(1/2)+I*2^(1/2), n = 1}-.585786438+1.414213562*I <- {T[2*n-1] = 2^(1/2)+I*2^(1/2), n = 2}-14.58578644+1.414213562*I <- {T[2*n-1] = 2^(1/2)+I*2^(1/2), n = 3}.414213562-1.414213562*I <- {T[2*n-1] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Successful |
| 24.15.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\frac{k!\StirlingnumberS@{n}{k}}{k+1}} | bernoulli(n)= sum((- 1)^(k)*(factorial(k)*Stirling2(n, k))/(k + 1), k = 0..n) |
BernoulliB[n]= Sum[(- 1)^(k)*Divide[(k)!*StirlingS2[n, k],k + 1], {k, 0, n}] |
Failure | Successful | Skip | - |
| 24.15.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{n} = \sum_{k=0}^{n}(-1)^{k}\binom{n+1}{k+1}\StirlingnumberS@{n+k}{k}\bigg{/}\binom{n+k}{k}} | bernoulli(n)= sum((- 1)^(k)*binomial(n + 1,k + 1)*Stirling2(n + k, k)/binomial(n + k,k), k = 0..n) |
BernoulliB[n]= Sum[(- 1)^(k)*Binomial[n + 1,k + 1]*StirlingS2[n + k, k]/Binomial[n + k,k], {k, 0, n}] |
Failure | Failure | Skip | Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BernoulliB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BernoulliB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}2.8284271247461903 <- {Rule[BernoulliB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}Complex[0.0, -2.8284271247461903] <- {Rule[BernoulliB[n], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.15.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=0}^{n}(-1)^{n+k}\Stirlingnumbers@{n+1}{k+1}\BernoullinumberB{k} = \frac{n!}{n+1}} | sum((- 1)^(n + k)* Stirling1(n + 1, k + 1)*bernoulli(k), k = 0..n)=(factorial(n))/(n + 1) |
Sum[(- 1)^(n + k)* StirlingS1[n + 1, k + 1]*BernoulliB[k], {k, 0, n}]=Divide[(n)!,n + 1] |
Failure | Failure | Skip | Successful |
| 24.16.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{t}{\ln@{1+t}} = \sum_{n=0}^{\infty}b_{n}t^{n}} | (t)/(ln(1 + t))= sum(b[n]*(t)^(n), n = 0..infinity) |
Divide[t,Log[1 + t]]= Sum[Subscript[b, n]*(t)^(n), {n, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 24.16.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \beta_{n}(\lambda) = n!b_{n}\lambda^{n}+\sum_{k=1}^{\floor{\ifrac{n}{2}}}\frac{n}{2k}\BernoullinumberB{2k}\Stirlingnumbers@{n-1}{2k-1}\lambda^{n-2k}} | beta[n]*(lambda)= factorial(n)*b[n]*(lambda)^(n)+ sum((n)/(2*k)*bernoulli(2*k)*Stirling1(n - 1, 2*k - 1)*(lambda)^(n - 2*k), k = 1..floor((n)/(2))) |
Subscript[\[Beta], n]*(\[Lambda])= (n)!*Subscript[b, n]*(\[Lambda])^(n)+ Sum[Divide[n,2*k]*BernoulliB[2*k]*StirlingS1[n - 1, 2*k - 1]*(\[Lambda])^(n - 2*k), {k, 1, Floor[Divide[n,2]]}] |
Failure | Failure | Skip | Skip |
| 24.16.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{x} = B_{n,\chi_{0}}(x-1)} | bernoulli(n, x)= B[n , chi[0]]*(x - 1) |
BernoulliB[n, x]= Subscript[B, n , Subscript[\[Chi], 0]]*(x - 1) |
Failure | Failure | Fail .5000000000 <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}.85786438e-1-1.414213562*I <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}-.328427124-2.828427124*I <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}.1666666667 <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Fail
0.5 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1], Rule[Subscript[χ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}0.16666666666666666 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1], Rule[Subscript[χ, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}0.5 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1], Rule[Subscript[χ, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}0.16666666666666666 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1], Rule[Subscript[χ, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.16.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n}@{x} = -\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1)} | euler(n, x)= -((2)^(1 - n))/(n + 1)*B[n + 1 , chi[4]]*(2*x - 1) |
EulerE[n, x]= -Divide[(2)^(1 - n),n + 1]*Subscript[B, n + 1 , Subscript[\[Chi], 4]]*(2*x - 1) |
Failure | Failure | Fail 1.207106781+.7071067810*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}3.621320343+2.121320343*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}6.035533905+3.535533905*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}.2357022604+.2357022604*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}... skip entries to safe data |
Successful |
| 24.19#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = \dfrac{N_{2n}}{D_{2n}}} | bernoulli(2*n)=(N[2*n])/(D[2*n]) |
BernoulliB[2*n]=Divide[Subscript[N, 2*n],Subscript[D, 2*n]] |
Failure | Failure | Fail -.8333333333 <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)+I*2^(1/2), n = 1}-1.033333333 <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)+I*2^(1/2), n = 2}-.9761904762 <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)+I*2^(1/2), n = 3}.1666666667+1.000000000*I <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, 2.414213562373095] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[2.414213562373095, 1.4142135623730951] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, 0.41421356237309515] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 24.19.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{t^{2}}{\cosh@@{t}-1} = -2\sum_{n=0}^{\infty}(2n-1)\BernoullinumberB{2n}\frac{t^{2n}}{(2n)!}} | ((t)^(2))/(cosh(t)- 1)= - 2*sum((2*n - 1)* bernoulli(2*n)*((t)^(2*n))/(factorial(2*n)), n = 0..infinity) |
Divide[(t)^(2),Cosh[t]- 1]= - 2*Sum[(2*n - 1)* BernoulliB[2*n]*Divide[(t)^(2*n),(2*n)!], {n, 0, Infinity}] |
Failure | Failure | Skip | Error |