# Results of Confluent Hypergeometric Functions

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DLMF Formula Maple Mathematica Symbolic
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Mathematica
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13.2.E1 ${\displaystyle{\displaystyle z\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(b-z% )\frac{\mathrm{d}w}{\mathrm{d}z}-aw=0}}$ z*diff(w, [z$(2)])+(b - z)* diff(w, z)- a*w = 0 z*D[w, {z, 2}]+(b - z)* D[w, z]- a*w = 0 Failure Failure Fail -0.-3.999999998*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)} -3.999999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)} -0.+3.999999998*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)} 3.999999998-0.*I <- {a = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data Fail Complex[0.0, -4.0] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} -4.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.0, 4.0] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} 4.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data 13.2.E2 ${\displaystyle{\displaystyle M\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{% \left(a\right)_{s}}}{{\left(b\right)_{s}}s!}z^{s}}}$ KummerM(a, b, z)= sum((pochhammer(a, s))/(pochhammer(b, s)*factorial(s))*(z)^(s), s = 0..infinity) Hypergeometric1F1[a, b, z]= Sum[Divide[Pochhammer[a, s],Pochhammer[b, s]*(s)!]*(z)^(s), {s, 0, Infinity}] Successful Successful - - 13.2.E3 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\sum_{s=0}^{\infty}% \frac{{\left(a\right)_{s}}}{\Gamma\left(b+s\right)s!}z^{s}}}$ KummerM(a, b, z)/GAMMA(b)= sum((pochhammer(a, s))/(GAMMA(b + s)*factorial(s))*(z)^(s), s = 0..infinity) Hypergeometric1F1Regularized[a, b, z]= Sum[Divide[Pochhammer[a, s],Gamma[b + s]*(s)!]*(z)^(s), {s, 0, Infinity}] Successful Successful - - 13.2.E4 ${\displaystyle{\displaystyle M\left(a,b,z\right)=\Gamma\left(b\right){\mathbf{% M}}\left(a,b,z\right)}}$ KummerM(a, b, z)= GAMMA(b)*KummerM(a, b, z)/GAMMA(b) Hypergeometric1F1[a, b, z]= Gamma[b]*Hypergeometric1F1Regularized[a, b, z] Successful Successful - - 13.2.E5 ${\displaystyle{\displaystyle\lim_{b\to-n}\frac{M\left(a,b,z\right)}{\Gamma% \left(b\right)}={\mathbf{M}}\left(a,-n,z\right)}}$ limit((KummerM(a, b, z))/(GAMMA(b)), b = - n)= KummerM(a, - n, z)/GAMMA(- n) Limit[Divide[Hypergeometric1F1[a, b, z],Gamma[b]], b -> - n]= Hypergeometric1F1Regularized[a, - n, z] Successful Successful - - 13.2.E5 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,-n,z\right)=\frac{{\left(a% \right)_{n+1}}}{(n+1)!}z^{n+1}M\left(a+n+1,n+2,z\right)}}$ KummerM(a, - n, z)/GAMMA(- n)=(pochhammer(a, n + 1))/(factorial(n + 1))*(z)^(n + 1)* KummerM(a + n + 1, n + 2, z) Hypergeometric1F1Regularized[a, - n, z]=Divide[Pochhammer[a, n + 1],(n + 1)!]*(z)^(n + 1)* Hypergeometric1F1[a + n + 1, n + 2, z] Failure Failure Fail Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} Float(undefined)+Float(undefined)*I <- {a = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data Successful 13.2.E7 ${\displaystyle{\displaystyle U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}% M\left(-m,b,z\right)}}$ KummerU(- m, b, z)=(- 1)^(m)* pochhammer(b, m)*KummerM(- m, b, z) HypergeometricU[- m, b, z]=(- 1)^(m)* Pochhammer[b, m]*Hypergeometric1F1[- m, b, z] Failure Failure Skip Successful 13.2.E7 ${\displaystyle{\displaystyle(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)=(% -1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(b+s\right)_{m-s}}(-z)% ^{s}}}$ (- 1)^(m)* pochhammer(b, m)*KummerM(- m, b, z)=(- 1)^(m)* sum(binomial(m,s)*pochhammer(b + s, m - s)*(- z)^(s), s = 0..m) (- 1)^(m)* Pochhammer[b, m]*Hypergeometric1F1[- m, b, z]=(- 1)^(m)* Sum[Binomial[m,s]*Pochhammer[b + s, m - s]*(- z)^(s), {s, 0, m}] Successful Successful - - 13.2.E8 ${\displaystyle{\displaystyle U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-% n\right)_{n}}}{z^{a+n}}M\left(-n,1-a-n,z\right)}}$ KummerU(a, a + n + 1, z)=((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))*KummerM(- n, 1 - a - n, z) HypergeometricU[a, a + n + 1, z]=Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]*Hypergeometric1F1[- n, 1 - a - n, z] Failure Failure Skip Successful 13.2.E8 ${\displaystyle{\displaystyle\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}M% \left(-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{s}{% \left(a\right)_{s}}z^{-s}}}$ ((- 1)^(n)* pochhammer(1 - a - n, n))/((z)^(a + n))*KummerM(- n, 1 - a - n, z)= (z)^(- a)* sum(binomial(n,s)*pochhammer(a, s)*(z)^(- s), s = 0..n) Divide[(- 1)^(n)* Pochhammer[1 - a - n, n],(z)^(a + n)]*Hypergeometric1F1[- n, 1 - a - n, z]= (z)^(- a)* Sum[Binomial[n,s]*Pochhammer[a, s]*(z)^(- s), {s, 0, n}] Failure Failure Skip Successful 13.2.E9 ${\displaystyle{\displaystyle U\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\Gamma% \left(a-n\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(n+1% \right)_{k}}k!}z^{k}\left(\ln z+\psi\left(a+k\right)-\psi\left(1+k\right)-\psi% \left(n+k+1\right)\right)+\frac{1}{\Gamma\left(a\right)}\sum_{k=1}^{n}\frac{(k% -1)!{\left(1-a+k\right)_{n-k}}}{(n-k)!}z^{-k}}}$ KummerU(a, n + 1, z)=((- 1)^(n + 1))/(factorial(n)*GAMMA(a - n))*sum((pochhammer(a, k))/(pochhammer(n + 1, k)*factorial(k))*(z)^(k)*(ln(z)+ Psi(a + k)- Psi(1 + k)- Psi(n + k + 1)), k = 0..infinity)+(1)/(GAMMA(a))*sum((factorial(k - 1)*pochhammer(1 - a + k, n - k))/(factorial(n - k))*(z)^(- k), k = 1..n) HypergeometricU[a, n + 1, z]=Divide[(- 1)^(n + 1),(n)!*Gamma[a - n]]*Sum[Divide[Pochhammer[a, k],Pochhammer[n + 1, k]*(k)!]*(z)^(k)*(Log[z]+ PolyGamma[a + k]- PolyGamma[1 + k]- PolyGamma[n + k + 1]), {k, 0, Infinity}]+Divide[1,Gamma[a]]*Sum[Divide[(k - 1)!*Pochhammer[1 - a + k, n - k],(n - k)!]*(z)^(- k), {k, 1, n}] Error Failure - Error 13.2.E10 ${\displaystyle{\displaystyle U\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_% {m}}M\left(-m,n+1,z\right)}}$ KummerU(- m, n + 1, z)=(- 1)^(m)* pochhammer(n + 1, m)*KummerM(- m, n + 1, z) HypergeometricU[- m, n + 1, z]=(- 1)^(m)* Pochhammer[n + 1, m]*Hypergeometric1F1[- m, n + 1, z] Failure Failure Successful Successful 13.2.E10 ${\displaystyle{\displaystyle(-1)^{m}{\left(n+1\right)_{m}}M\left(-m,n+1,z% \right)=(-1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(n+s+1\right)% _{m-s}}(-z)^{s}}}$ (- 1)^(m)* pochhammer(n + 1, m)*KummerM(- m, n + 1, z)=(- 1)^(m)* sum(binomial(m,s)*pochhammer(n + s + 1, m - s)*(- z)^(s), s = 0..m) (- 1)^(m)* Pochhammer[n + 1, m]*Hypergeometric1F1[- m, n + 1, z]=(- 1)^(m)* Sum[Binomial[m,s]*Pochhammer[n + s + 1, m - s]*(- z)^(s), {s, 0, m}] Successful Successful - - 13.2.E11 ${\displaystyle{\displaystyle U\left(a,-n,z\right)=z^{n+1}U\left(a+n+1,n+2,z% \right)}}$ KummerU(a, - n, z)= (z)^(n + 1)* KummerU(a + n + 1, n + 2, z) HypergeometricU[a, - n, z]= (z)^(n + 1)* HypergeometricU[a + n + 1, n + 2, z] Successful Successful - - 13.2.E12 ${\displaystyle{\displaystyle U\left(a,b,ze^{2\pi\mathrm{i}m}\right)=\frac{2\pi% \mathrm{i}e^{-\pi\mathrm{i}bm}\sin\left(\pi bm\right)}{\Gamma\left(1+a-b\right% )\sin\left(\pi b\right)}{\mathbf{M}}\left(a,b,z\right)+e^{-2\pi\mathrm{i}bm}U% \left(a,b,z\right)}}$ KummerU(a, b, z*exp(2*Pi*I*m))=(2*Pi*I*exp(- Pi*I*b*m)*sin(Pi*b*m))/(GAMMA(1 + a - b)*sin(Pi*b))*KummerM(a, b, z)/GAMMA(b)+ exp(- 2*Pi*I*b*m)*KummerU(a, b, z) HypergeometricU[a, b, z*Exp[2*Pi*I*m]]=Divide[2*Pi*I*Exp[- Pi*I*b*m]*Sin[Pi*b*m],Gamma[1 + a - b]*Sin[Pi*b]]*Hypergeometric1F1Regularized[a, b, z]+ Exp[- 2*Pi*I*b*m]*HypergeometricU[a, b, z] Failure Failure Fail 584.8702437+1098.665595*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1} 448650.07-8984458.84*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2} -.361175805e11+.540703722e11*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3} 1655.171849-5530.515123*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1} ... skip entries to safe data Skip 13.2.E33 ${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),z% ^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right)\right\}=\sin\left(\pi b\right)z^{-b% }e^{z}/\pi}}$ (KummerM(a, b, z)/GAMMA(b))*diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))= sin(Pi*b)*(z)^(- b)* exp(z)/ Pi Wronskian[{Hypergeometric1F1Regularized[a, b, z], (z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z]}, z]= Sin[Pi*b]*(z)^(- b)* Exp[z]/ Pi Failure Failure Successful Skip 13.2.E34 ${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),U% \left(a,b,z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a\right)}}}$ (KummerM(a, b, z)/GAMMA(b))*diff(KummerU(a, b, z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(KummerU(a, b, z))= -((z)^(- b)* exp(z))/(GAMMA(a)) Wronskian[{Hypergeometric1F1Regularized[a, b, z], HypergeometricU[a, b, z]}, z]= -Divide[(z)^(- b)* Exp[z],Gamma[a]] Failure Failure Successful Skip 13.2.E35 ${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e% ^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=\ifrac{e^{-b\pi\mathrm{i}}% z^{-b}e^{z}}{\Gamma\left(b-a\right)}}}$ (KummerM(a, b, z)/GAMMA(b))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z))=(exp(- b*Pi*I)*(z)^(- b)* exp(z))/(GAMMA(b - a)) Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z]=Divide[Exp[- b*Pi*I]*(z)^(- b)* Exp[z],Gamma[b - a]] Failure Failure Skip Skip 13.2.E35 ${\displaystyle{\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e% ^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=\ifrac{e^{+b\pi\mathrm{i}}% z^{-b}e^{z}}{\Gamma\left(b-a\right)}}}$ (KummerM(a, b, z)/GAMMA(b))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff(KummerM(a, b, z)/GAMMA(b), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z))=(exp(+ b*Pi*I)*(z)^(- b)* exp(z))/(GAMMA(b - a)) Wronskian[{Hypergeometric1F1Regularized[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z]=Divide[Exp[+ b*Pi*I]*(z)^(- b)* Exp[z],Gamma[b - a]] Failure Failure Skip Skip 13.2.E36 ${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2% -b,z\right),U\left(a,b,z\right)\right\}=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a-b+1% \right)}}}$ ((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(KummerU(a, b, z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(KummerU(a, b, z))= -((z)^(- b)* exp(z))/(GAMMA(a - b + 1)) Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], HypergeometricU[a, b, z]}, z]= -Divide[(z)^(- b)* Exp[z],Gamma[a - b + 1]] Failure Failure Skip Skip 13.2.E37 ${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2% -b,z\right),e^{z}U\left(b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=-\ifrac{z^{-b% }e^{z}}{\Gamma\left(1-a\right)}}}$ ((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z))= -((z)^(- b)* exp(z))/(GAMMA(1 - a)) Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z]= -Divide[(z)^(- b)* Exp[z],Gamma[1 - a]] Failure Failure Skip Successful 13.2.E37 ${\displaystyle{\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2% -b,z\right),e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=-\ifrac{z^{-b% }e^{z}}{\Gamma\left(1-a\right)}}}$ ((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff((z)^(1 - b)* KummerM(a - b + 1, 2 - b, z)/GAMMA(2 - b), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z))= -((z)^(- b)* exp(z))/(GAMMA(1 - a)) Wronskian[{(z)^(1 - b)* Hypergeometric1F1Regularized[a - b + 1, 2 - b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z]= -Divide[(z)^(- b)* Exp[z],Gamma[1 - a]] Failure Failure Skip Skip 13.2.E38 ${\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(% b-a,b,e^{+\pi\mathrm{i}}z\right)\right\}=e^{+(a-b)\pi\mathrm{i}}z^{-b}e^{z}}}$ (KummerU(a, b, z))*diff(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z), z)-diff(KummerU(a, b, z), z)*(exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z))= exp(+(a - b)* Pi*I)*(z)^(- b)* exp(z) Wronskian[{HypergeometricU[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z]}, z]= Exp[+(a - b)* Pi*I]*(z)^(- b)* Exp[z] Failure Failure Skip Fail Complex[1040.14465936905, 3523.550863963589] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[13.933478379950422, -18.985981055998398] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} Complex[16167.755810004226, 20483.57845334895] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[-167.2507901552425, 2.9337620233109254] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data 13.2.E38 ${\displaystyle{\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(% b-a,b,e^{-\pi\mathrm{i}}z\right)\right\}=e^{-(a-b)\pi\mathrm{i}}z^{-b}e^{z}}}$ (KummerU(a, b, z))*diff(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z), z)-diff(KummerU(a, b, z), z)*(exp(z)*KummerU(b - a, b, exp(- Pi*I)*z))= exp(-(a - b)* Pi*I)*(z)^(- b)* exp(z) Wronskian[{HypergeometricU[a, b, z], Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z]}, z]= Exp[-(a - b)* Pi*I]*(z)^(- b)* Exp[z] Failure Failure Skip Fail Complex[-26409.287510504182, -21215.250458979182] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[17917.63845480152, -4449.098851771366] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[128856.58558615872, -203204.6357206061] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[-130654.53246573739, 11199.95676626326] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[b, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data 13.2.E39 ${\displaystyle{\displaystyle M\left(a,b,z\right)=e^{z}M\left(b-a,b,-z\right)}}$ KummerM(a, b, z)= exp(z)*KummerM(b - a, b, - z) Hypergeometric1F1[a, b, z]= Exp[z]*Hypergeometric1F1[b - a, b, - z] Failure Successful Successful - 13.2.E40 ${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{1-b}U\left(a-b+1,2-b,z% \right)}}$ KummerU(a, b, z)= (z)^(1 - b)* KummerU(a - b + 1, 2 - b, z) HypergeometricU[a, b, z]= (z)^(1 - b)* HypergeometricU[a - b + 1, 2 - b, z] Successful Successful - - 13.2.E41 ${\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=% \frac{e^{-a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^% {+(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{+\pi\mathrm{i% }}z\right)}}$ (1)/(GAMMA(b))*KummerM(a, b, z)=(exp(- a*Pi*I))/(GAMMA(b - a))*KummerU(a, b, z)+(exp(+(b - a)* Pi*I))/(GAMMA(a))*exp(z)*KummerU(b - a, b, exp(+ Pi*I)*z) Divide[1,Gamma[b]]*Hypergeometric1F1[a, b, z]=Divide[Exp[- a*Pi*I],Gamma[b - a]]*HypergeometricU[a, b, z]+Divide[Exp[+(b - a)* Pi*I],Gamma[a]]*Exp[z]*HypergeometricU[b - a, b, Exp[+ Pi*I]*z] Failure Failure Fail 17637856.16+44349536.15*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} 78404.04567+70170.88583*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} 23503366.51-739194412.4*I <- {a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)} -413147.5251+1810381.777*I <- {a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)} ... skip entries to safe data Skip 13.2.E41 ${\displaystyle{\displaystyle\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=% \frac{e^{+a\pi\mathrm{i}}}{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^% {-(b-a)\pi\mathrm{i}}}{\Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{-\pi\mathrm{i% }}z\right)}}$ (1)/(GAMMA(b))*KummerM(a, b, z)=(exp(+ a*Pi*I))/(GAMMA(b - a))*KummerU(a, b, z)+(exp(-(b - a)* Pi*I))/(GAMMA(a))*exp(z)*KummerU(b - a, b, exp(- Pi*I)*z) Divide[1,Gamma[b]]*Hypergeometric1F1[a, b, z]=Divide[Exp[+ a*Pi*I],Gamma[b - a]]*HypergeometricU[a, b, z]+Divide[Exp[-(b - a)* Pi*I],Gamma[a]]*Exp[z]*HypergeometricU[b - a, b, Exp[- Pi*I]*z] Failure Failure Fail 8.816149469+15.35727015*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 3.036467728-4.734652938*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} 237.2244957-69.52948040*I <- {a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)} 40.35920508+88.71475163*I <- {a = 2^(1/2)+I*2^(1/2), b = -2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)} ... skip entries to safe data Skip 13.2.E42 ${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{\Gamma\left(1-b\right)}% {\Gamma\left(a-b+1\right)}M\left(a,b,z\right)+\frac{\Gamma\left(b-1\right)}{% \Gamma\left(a\right)}z^{1-b}M\left(a-b+1,2-b,z\right)}}$ KummerU(a, b, z)=(GAMMA(1 - b))/(GAMMA(a - b + 1))*KummerM(a, b, z)+(GAMMA(b - 1))/(GAMMA(a))*(z)^(1 - b)* KummerM(a - b + 1, 2 - b, z) HypergeometricU[a, b, z]=Divide[Gamma[1 - b],Gamma[a - b + 1]]*Hypergeometric1F1[a, b, z]+Divide[Gamma[b - 1],Gamma[a]]*(z)^(1 - b)* Hypergeometric1F1[a - b + 1, 2 - b, z] Successful Successful - - 13.3.E1 ${\displaystyle{\displaystyle(b-a)M\left(a-1,b,z\right)+(2a-b+z)M\left(a,b,z% \right)-aM\left(a+1,b,z\right)=0}}$ (b - a)* KummerM(a - 1, b, z)+(2*a - b + z)* KummerM(a, b, z)- a*KummerM(a + 1, b, z)= 0 (b - a)* Hypergeometric1F1[a - 1, b, z]+(2*a - b + z)* Hypergeometric1F1[a, b, z]- a*Hypergeometric1F1[a + 1, b, z]= 0 Successful Successful - - 13.3.E2 ${\displaystyle{\displaystyle b(b-1)M\left(a,b-1,z\right)+b(1-b-z)M\left(a,b,z% \right)+z(b-a)M\left(a,b+1,z\right)=0}}$ b*(b - 1)* KummerM(a, b - 1, z)+ b*(1 - b - z)* KummerM(a, b, z)+ z*(b - a)* KummerM(a, b + 1, z)= 0 b*(b - 1)* Hypergeometric1F1[a, b - 1, z]+ b*(1 - b - z)* Hypergeometric1F1[a, b, z]+ z*(b - a)* Hypergeometric1F1[a, b + 1, z]= 0 Successful Successful - - 13.3.E3 ${\displaystyle{\displaystyle(a-b+1)M\left(a,b,z\right)-aM\left(a+1,b,z\right)+% (b-1)M\left(a,b-1,z\right)=0}}$ (a - b + 1)* KummerM(a, b, z)- a*KummerM(a + 1, b, z)+(b - 1)* KummerM(a, b - 1, z)= 0 (a - b + 1)* Hypergeometric1F1[a, b, z]- a*Hypergeometric1F1[a + 1, b, z]+(b - 1)* Hypergeometric1F1[a, b - 1, z]= 0 Successful Successful - - 13.3.E4 ${\displaystyle{\displaystyle bM\left(a,b,z\right)-bM\left(a-1,b,z\right)-zM% \left(a,b+1,z\right)=0}}$ b*KummerM(a, b, z)- b*KummerM(a - 1, b, z)- z*KummerM(a, b + 1, z)= 0 b*Hypergeometric1F1[a, b, z]- b*Hypergeometric1F1[a - 1, b, z]- z*Hypergeometric1F1[a, b + 1, z]= 0 Successful Successful - - 13.3.E5 ${\displaystyle{\displaystyle b(a+z)M\left(a,b,z\right)+z(a-b)M\left(a,b+1,z% \right)-abM\left(a+1,b,z\right)=0}}$ b*(a + z)* KummerM(a, b, z)+ z*(a - b)* KummerM(a, b + 1, z)- a*b*KummerM(a + 1, b, z)= 0 b*(a + z)* Hypergeometric1F1[a, b, z]+ z*(a - b)* Hypergeometric1F1[a, b + 1, z]- a*b*Hypergeometric1F1[a + 1, b, z]= 0 Successful Successful - - 13.3.E6 ${\displaystyle{\displaystyle(a-1+z)M\left(a,b,z\right)+(b-a)M\left(a-1,b,z% \right)+(1-b)M\left(a,b-1,z\right)=0}}$ (a - 1 + z)* KummerM(a, b, z)+(b - a)* KummerM(a - 1, b, z)+(1 - b)* KummerM(a, b - 1, z)= 0 (a - 1 + z)* Hypergeometric1F1[a, b, z]+(b - a)* Hypergeometric1F1[a - 1, b, z]+(1 - b)* Hypergeometric1F1[a, b - 1, z]= 0 Successful Successful - - 13.3.E7 ${\displaystyle{\displaystyle U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)% +a(a-b+1)U\left(a+1,b,z\right)=0}}$ KummerU(a - 1, b, z)+(b - 2*a - z)* KummerU(a, b, z)+ a*(a - b + 1)* KummerU(a + 1, b, z)= 0 HypergeometricU[a - 1, b, z]+(b - 2*a - z)* HypergeometricU[a, b, z]+ a*(a - b + 1)* HypergeometricU[a + 1, b, z]= 0 Successful Successful - - 13.3.E8 ${\displaystyle{\displaystyle(b-a-1)U\left(a,b-1,z\right)+(1-b-z)U\left(a,b,z% \right)+zU\left(a,b+1,z\right)=0}}$ (b - a - 1)* KummerU(a, b - 1, z)+(1 - b - z)* KummerU(a, b, z)+ z*KummerU(a, b + 1, z)= 0 (b - a - 1)* HypergeometricU[a, b - 1, z]+(1 - b - z)* HypergeometricU[a, b, z]+ z*HypergeometricU[a, b + 1, z]= 0 Successful Successful - - 13.3.E9 ${\displaystyle{\displaystyle U\left(a,b,z\right)-aU\left(a+1,b,z\right)-U\left% (a,b-1,z\right)=0}}$ KummerU(a, b, z)- a*KummerU(a + 1, b, z)- KummerU(a, b - 1, z)= 0 HypergeometricU[a, b, z]- a*HypergeometricU[a + 1, b, z]- HypergeometricU[a, b - 1, z]= 0 Successful Successful - - 13.3.E10 ${\displaystyle{\displaystyle(b-a)U\left(a,b,z\right)+U\left(a-1,b,z\right)-zU% \left(a,b+1,z\right)=0}}$ (b - a)* KummerU(a, b, z)+ KummerU(a - 1, b, z)- z*KummerU(a, b + 1, z)= 0 (b - a)* HypergeometricU[a, b, z]+ HypergeometricU[a - 1, b, z]- z*HypergeometricU[a, b + 1, z]= 0 Successful Successful - - 13.3.E11 ${\displaystyle{\displaystyle(a+z)U\left(a,b,z\right)-zU\left(a,b+1,z\right)+a(% b-a-1)U\left(a+1,b,z\right)=0}}$ (a + z)* KummerU(a, b, z)- z*KummerU(a, b + 1, z)+ a*(b - a - 1)* KummerU(a + 1, b, z)= 0 (a + z)* HypergeometricU[a, b, z]- z*HypergeometricU[a, b + 1, z]+ a*(b - a - 1)* HypergeometricU[a + 1, b, z]= 0 Successful Successful - - 13.3.E12 ${\displaystyle{\displaystyle(a-1+z)U\left(a,b,z\right)-U\left(a-1,b,z\right)+(% a-b+1)U\left(a,b-1,z\right)=0}}$ (a - 1 + z)* KummerU(a, b, z)- KummerU(a - 1, b, z)+(a - b + 1)* KummerU(a, b - 1, z)= 0 (a - 1 + z)* HypergeometricU[a, b, z]- HypergeometricU[a - 1, b, z]+(a - b + 1)* HypergeometricU[a, b - 1, z]= 0 Successful Successful - - 13.3.E13 ${\displaystyle{\displaystyle(a+1)zM\left(a+2,b+2,z\right)+(b+1)(b-z)M\left(a+1% ,b+1,z\right)-b(b+1)M\left(a,b,z\right)=0}}$ (a + 1)* z*KummerM(a + 2, b + 2, z)+(b + 1)*(b - z)* KummerM(a + 1, b + 1, z)- b*(b + 1)* KummerM(a, b, z)= 0 (a + 1)* z*Hypergeometric1F1[a + 2, b + 2, z]+(b + 1)*(b - z)* Hypergeometric1F1[a + 1, b + 1, z]- b*(b + 1)* Hypergeometric1F1[a, b, z]= 0 Successful Successful - - 13.3.E14 ${\displaystyle{\displaystyle(a+1)zU\left(a+2,b+2,z\right)+(z-b)U\left(a+1,b+1,% z\right)-U\left(a,b,z\right)=0}}$ (a + 1)* z*KummerU(a + 2, b + 2, z)+(z - b)* KummerU(a + 1, b + 1, z)- KummerU(a, b, z)= 0 (a + 1)* z*HypergeometricU[a + 2, b + 2, z]+(z - b)* HypergeometricU[a + 1, b + 1, z]- HypergeometricU[a, b, z]= 0 Successful Successful - - 13.3.E15 ${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}M\left(a,b,z\right)=% \frac{a}{b}M\left(a+1,b+1,z\right)}}$ diff(KummerM(a, b, z), z)=(a)/(b)*KummerM(a + 1, b + 1, z) D[Hypergeometric1F1[a, b, z], z]=Divide[a,b]*Hypergeometric1F1[a + 1, b + 1, z] Successful Successful - - 13.3.E16 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}M\left(a% ,b,z\right)=\frac{{\left(a\right)_{n}}}{{\left(b\right)_{n}}}M\left(a+n,b+n,z% \right)}}$ diff(KummerM(a, b, z), [z$(n)])=(pochhammer(a, n))/(pochhammer(b, n))*KummerM(a + n, b + n, z) D[Hypergeometric1F1[a, b, z], {z, n}]=Divide[Pochhammer[a, n],Pochhammer[b, n]]*Hypergeometric1F1[a + n, b + n, z] Successful Failure - Skip
13.3.E17 ${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{a-1}M\left(a,b,z\right)\right)={\left(a\right)_{n}}z^{a+n-1}M\left(a+% n,b,z\right)}}$ (z*diff(z, z))^(n)*((z)^(a - 1)* KummerM(a, b, z))= pochhammer(a, n)*(z)^(a + n - 1)* KummerM(a + n, b, z) (z*D[z, z])^(n)*((z)^(a - 1)* Hypergeometric1F1[a, b, z])= Pochhammer[a, n]*(z)^(a + n - 1)* Hypergeometric1F1[a + n, b, z] Failure Failure
Fail
2.537884887+11.89104377*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1}
-123.7627467+81.19826795*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2}
-1555.783365-1131.870657*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3}
-1.589608076+60.84364464*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
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13.3.E18 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}M\left(a,b,z\right)\right)={\left(b-n\right)_{n}}z^{b-n-1}M\left(a,b-n,z% \right)}}$ diff((z)^(b - 1)* KummerM(a, b, z), [z$(n)])= pochhammer(b - n, n)*(z)^(b - n - 1)* KummerM(a, b - n, z) D[(z)^(b - 1)* Hypergeometric1F1[a, b, z], {z, n}]= Pochhammer[b - n, n]*(z)^(b - n - 1)* Hypergeometric1F1[a, b - n, z] Failure Failure Successful Skip 13.3.E19 ${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{b-a-1}e^{-z}M\left(a,b,z\right)\right)={\left(b-a\right)_{n}}z^{b-a+n% -1}e^{-z}M\left(a-n,b,z\right)}}$ (z*diff(z, z))^(n)*((z)^(b - a - 1)* exp(- z)*KummerM(a, b, z))= pochhammer(b - a, n)*(z)^(b - a + n - 1)* exp(- z)*KummerM(a - n, b, z) (z*D[z, z])^(n)*((z)^(b - a - 1)* Exp[- z]*Hypergeometric1F1[a, b, z])= Pochhammer[b - a, n]*(z)^(b - a + n - 1)* Exp[- z]*Hypergeometric1F1[a - n, b, z] Failure Failure Fail 1.000000000+0.*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} 1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} 0.+3.999999998*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} 1.000000000+0.*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data Skip 13.3.E20 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-z}M\left(a,b,z\right)\right)=(-1)^{n}\frac{{\left(b-a\right)_{n}}}{{\left(b% \right)_{n}}}e^{-z}M\left(a,b+n,z\right)}}$ diff(exp(- z)*KummerM(a, b, z), [z$(n)])=(- 1)^(n)*(pochhammer(b - a, n))/(pochhammer(b, n))*exp(- z)*KummerM(a, b + n, z) D[Exp[- z]*Hypergeometric1F1[a, b, z], {z, n}]=(- 1)^(n)*Divide[Pochhammer[b - a, n],Pochhammer[b, n]]*Exp[- z]*Hypergeometric1F1[a, b + n, z] Failure Failure Successful Skip
13.3.E21 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}e^{-z}M\left(a,b,z\right)\right)={\left(b-n\right)_{n}}z^{b-n-1}e^{-z}M% \left(a-n,b-n,z\right)}}$ diff((z)^(b - 1)* exp(- z)*KummerM(a, b, z), [z$(n)])= pochhammer(b - n, n)*(z)^(b - n - 1)* exp(- z)*KummerM(a - n, b - n, z) D[(z)^(b - 1)* Exp[- z]*Hypergeometric1F1[a, b, z], {z, n}]= Pochhammer[b - n, n]*(z)^(b - n - 1)* Exp[- z]*Hypergeometric1F1[a - n, b - n, z] Failure Failure Skip Error 13.3.E22 ${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}U\left(a,b,z\right)=% -aU\left(a+1,b+1,z\right)}}$ diff(KummerU(a, b, z), z)= - a*KummerU(a + 1, b + 1, z) D[HypergeometricU[a, b, z], z]= - a*HypergeometricU[a + 1, b + 1, z] Successful Successful - - 13.3.E23 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}U\left(a% ,b,z\right)=(-1)^{n}{\left(a\right)_{n}}U\left(a+n,b+n,z\right)}}$ diff(KummerU(a, b, z), [z$(n)])=(- 1)^(n)* pochhammer(a, n)*KummerU(a + n, b + n, z) D[HypergeometricU[a, b, z], {z, n}]=(- 1)^(n)* Pochhammer[a, n]*HypergeometricU[a + n, b + n, z] Failure Successful Skip -
13.3.E24 ${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{a-1}U\left(a,b,z\right)\right)={\left(a\right)_{n}}{\left(a-b+1\right% )_{n}}z^{a+n-1}U\left(a+n,b,z\right)}}$ (z*diff(z, z))^(n)*((z)^(a - 1)* KummerU(a, b, z))= pochhammer(a, n)*pochhammer(a - b + 1, n)*(z)^(a + n - 1)* KummerU(a + n, b, z) (z*D[z, z])^(n)*((z)^(a - 1)* HypergeometricU[a, b, z])= Pochhammer[a, n]*Pochhammer[a - b + 1, n]*(z)^(a + n - 1)* HypergeometricU[a + n, b, z] Failure Failure
Fail
.3178044521-.5812355890e-1*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1}
.5638915996+.3833395878*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2}
.2833587160+.898459259*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3}
.2659178351-.5754539144*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
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13.3.E25 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}U\left(a,b,z\right)\right)=(-1)^{n}{\left(a-b+1\right)_{n}}z^{b-n-1}U% \left(a,b-n,z\right)}}$ diff((z)^(b - 1)* KummerU(a, b, z), [z$(n)])=(- 1)^(n)* pochhammer(a - b + 1, n)*(z)^(b - n - 1)* KummerU(a, b - n, z) D[(z)^(b - 1)* HypergeometricU[a, b, z], {z, n}]=(- 1)^(n)* Pochhammer[a - b + 1, n]*(z)^(b - n - 1)* HypergeometricU[a, b - n, z] Failure Failure Skip Skip 13.3.E26 ${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{b-a-1}e^{-z}U\left(a,b,z\right)\right)=(-1)^{n}z^{b-a+n-1}e^{-z}U% \left(a-n,b,z\right)}}$ (z*diff(z, z))^(n)*((z)^(b - a - 1)* exp(- z)*KummerU(a, b, z))=(- 1)^(n)* (z)^(b - a + n - 1)* exp(- z)*KummerU(a - n, b, z) (z*D[z, z])^(n)*((z)^(b - a - 1)* Exp[- z]*HypergeometricU[a, b, z])=(- 1)^(n)* (z)^(b - a + n - 1)* Exp[- z]*HypergeometricU[a - n, b, z] Failure Failure Fail -.6426838098-.1638932643*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1} -2.885602225+1.867279788*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2} -8.024434137+19.17405510*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3} .5784828818e-1+.5986041895e-1*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1} ... skip entries to safe data Skip 13.3.E27 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-z}U\left(a,b,z\right)\right)=(-1)^{n}e^{-z}U\left(a,b+n,z\right)}}$ diff(exp(- z)*KummerU(a, b, z), [z$(n)])=(- 1)^(n)* exp(- z)*KummerU(a, b + n, z) D[Exp[- z]*HypergeometricU[a, b, z], {z, n}]=(- 1)^(n)* Exp[- z]*HypergeometricU[a, b + n, z] Failure Failure Skip Skip
13.3.E28 ${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {b-1}e^{-z}U\left(a,b,z\right)\right)=(-1)^{n}z^{b-n-1}e^{-z}U\left(a-n,b-n,z% \right)}}$ diff((z)^(b - 1)* exp(- z)*KummerU(a, b, z), [z$(n)])=(- 1)^(n)* (z)^(b - n - 1)* exp(- z)*KummerU(a - n, b - n, z) D[(z)^(b - 1)* Exp[- z]*HypergeometricU[a, b, z], {z, n}]=(- 1)^(n)* (z)^(b - n - 1)* Exp[- z]*HypergeometricU[a - n, b - n, z] Error Failure - Error 13.3.E29 ${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=% z^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{n}}}$ (z*diff(z, z))^(n)= (z)^(n)* diff((z)^(n), [z$(n)]) (z*D[z, z])^(n)= (z)^(n)* D[(z)^(n), {z, n}] Failure Failure
Fail
28.28427122-28.28427122*I <- {z = 2^(1/2)+I*2^(1/2), n = 3}
28.28427122+28.28427122*I <- {z = 2^(1/2)-I*2^(1/2), n = 3}
-28.28427122+28.28427122*I <- {z = -2^(1/2)-I*2^(1/2), n = 3}
-28.28427122-28.28427122*I <- {z = -2^(1/2)+I*2^(1/2), n = 3}
Fail
Complex[28.284271247461902, -28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[28.284271247461902, 28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-28.284271247461902, 28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-28.284271247461902, -28.284271247461902] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
13.4.E1 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(a\right)\Gamma\left(b-a\right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}% \mathrm{d}t}}$ KummerM(a, b, z)/GAMMA(b)=(1)/(GAMMA(a)*GAMMA(b - a))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = 0..1) Hypergeometric1F1Regularized[a, b, z]=Divide[1,Gamma[a]*Gamma[b - a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, 0, 1}] Successful Failure - Skip
13.4.E2 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma% \left(b-c\right)}\int_{0}^{1}{\mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-% 1}\mathrm{d}t}}$ KummerM(a, b, z)/GAMMA(b)=(1)/(GAMMA(b - c))*int(KummerM(a, c, z*t)/GAMMA(c)*(t)^(c - 1)*(1 - t)^(b - c - 1), t = 0..1) Hypergeometric1F1Regularized[a, b, z]=Divide[1,Gamma[b - c]]*Integrate[Hypergeometric1F1Regularized[a, c, z*t]*(t)^(c - 1)*(1 - t)^(b - c - 1), {t, 0, 1}] Successful Failure - Skip
13.4.E3 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{% 2}-\frac{1}{2}b}}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}% b-\frac{1}{2}}J_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ KummerM(a, b, - z)/GAMMA(b)=((z)^((1)/(2)-(1)/(2)*b))/(GAMMA(a))*int(exp(- t)*(t)^(a -(1)/(2)*b -(1)/(2))* BesselJ(b - 1, 2*sqrt(z*t)), t = 0..infinity) Hypergeometric1F1Regularized[a, b, - z]=Divide[(z)^(Divide[1,2]-Divide[1,2]*b),Gamma[a]]*Integrate[Exp[- t]*(t)^(a -Divide[1,2]*b -Divide[1,2])* BesselJ[b - 1, 2*Sqrt[z*t]], {t, 0, Infinity}] Failure Failure Skip Error
13.4.E4 ${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)% }\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\mathrm{d}t}}$ KummerU(a, b, z)=(1)/(GAMMA(a))*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = 0..infinity) HypergeometricU[a, b, z]=Divide[1,Gamma[a]]*Integrate[Exp[- z*t]*(t)^(a - 1)*(1 + t)^(b - a - 1), {t, 0, Infinity}] Successful Failure - Error
13.4.E5 ${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a% \right)\Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^% {-t}t^{a-1}}{t+z}\mathrm{d}t}}$ KummerU(a, b, z)=((z)^(1 - a))/(GAMMA(a)*GAMMA(1 + a - b))*int((KummerU(b - a, b, t)*exp(- t)*(t)^(a - 1))/(t + z), t = 0..infinity) HypergeometricU[a, b, z]=Divide[(z)^(1 - a),Gamma[a]*Gamma[1 + a - b]]*Integrate[Divide[HypergeometricU[b - a, b, t]*Exp[- t]*(t)^(a - 1),t + z], {t, 0, Infinity}] Failure Failure Skip Error
13.4.E6 ${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{% \Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t% \right)e^{-t}t^{b+n-1}}{t+z}\mathrm{d}t}}$ KummerU(a, b, z)=((- 1)^(n)* (z)^(1 - b - n))/(GAMMA(1 + a - b))*int((KummerM(b - a, b, t)/GAMMA(b)*exp(- t)*(t)^(b + n - 1))/(t + z), t = 0..infinity) HypergeometricU[a, b, z]=Divide[(- 1)^(n)* (z)^(1 - b - n),Gamma[1 + a - b]]*Integrate[Divide[Hypergeometric1F1Regularized[b - a, b, t]*Exp[- t]*(t)^(b + n - 1),t + z], {t, 0, Infinity}] Failure Failure Skip Error
13.4.E7 ${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1% }{2}b}}{\Gamma\left(a\right)\Gamma\left(a-b+1\right)}\*\int_{0}^{\infty}e^{-t}% t^{a-\frac{1}{2}b-\frac{1}{2}}K_{b-1}\left(2\sqrt{zt}\right)\mathrm{d}t}}$ KummerU(a, b, z)=(2*(z)^((1)/(2)-(1)/(2)*b))/(GAMMA(a)*GAMMA(a - b + 1))* int(exp(- t)*(t)^(a -(1)/(2)*b -(1)/(2))* BesselK(b - 1, 2*sqrt(z*t)), t = 0..infinity) HypergeometricU[a, b, z]=Divide[2*(z)^(Divide[1,2]-Divide[1,2]*b),Gamma[a]*Gamma[a - b + 1]]* Integrate[Exp[- t]*(t)^(a -Divide[1,2]*b -Divide[1,2])* BesselK[b - 1, 2*Sqrt[z*t]], {t, 0, Infinity}] Successful Failure - Error
13.4.E8 ${\displaystyle{\displaystyle U\left(a,b,z\right)=z^{c-a}\*\int_{0}^{\infty}e^{% -zt}t^{c-1}{{}_{2}{\mathbf{F}}_{1}}\left(a,a-b+1;c;-t\right)\mathrm{d}t}}$ KummerU(a, b, z)= (z)^(c - a)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([a , a - b + 1], [c], - t), t = 0..infinity) HypergeometricU[a, b, z]= (z)^(c - a)* Integrate[Exp[- z*t]*(t)^(c - 1)* HypergeometricPFQRegularized[{a , a - b + 1}, {c}, - t], {t, 0, Infinity}] Failure Failure Skip Error
13.4.E9 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1% +a-b\right)}{2\pi\mathrm{i}\Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(% t-1)^{b-a-1}}\mathrm{d}t}}$ KummerM(a, b, z)/GAMMA(b)=(GAMMA(1 + a - b))/(2*Pi*I*GAMMA(a))*int(exp(z*t)*(t)^(a - 1)*(t - 1)^(b - a - 1), t = 0..(1 +)) Hypergeometric1F1Regularized[a, b, z]=Divide[Gamma[1 + a - b],2*Pi*I*Gamma[a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(t - 1)^(b - a - 1), {t, 0, (1 +)}] Error Failure - Error
13.4.E10 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-a\pi\mathrm{i}}% \frac{\Gamma\left(1-a\right)}{2\pi\mathrm{i}\Gamma\left(b-a\right)}\int_{1}^{(% 0+)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}$ KummerM(a, b, z)/GAMMA(b)= exp(- a*Pi*I)*(GAMMA(1 - a))/(2*Pi*I*GAMMA(b - a))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = 1..(0 +)) Hypergeometric1F1Regularized[a, b, z]= Exp[- a*Pi*I]*Divide[Gamma[1 - a],2*Pi*I*Gamma[b - a]]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, 1, (0 +)}] Error Failure - Error
13.4.E11 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=e^{-b\pi\mathrm{i}}% \Gamma\left(1-a\right)\Gamma\left(1+a-b\right)\*\frac{1}{4\pi^{2}}\int_{\alpha% }^{(0+,1+,0-,1-)}e^{zt}t^{a-1}{(1-t)^{b-a-1}}\mathrm{d}t}}$ KummerM(a, b, z)/GAMMA(b)= exp(- b*Pi*I)*GAMMA(1 - a)*GAMMA(1 + a - b)*(1)/(4*(Pi)^(2))*int(exp(z*t)*(t)^(a - 1)*(1 - t)^(b - a - 1), t = alpha..(0 + , 1 + , 0 - , 1 -)) Hypergeometric1F1Regularized[a, b, z]= Exp[- b*Pi*I]*Gamma[1 - a]*Gamma[1 + a - b]*Divide[1,4*(Pi)^(2)]*Integrate[Exp[z*t]*(t)^(a - 1)*(1 - t)^(b - a - 1), {t, \[Alpha], (0 + , 1 + , 0 - , 1 -)}] Error Failure - Error
13.4.E12 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b% \right)}{2\pi\mathrm{i}}z^{1-b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{% \mathbf{F}}_{1}}\left(a,b;c;\ifrac{1}{t}\right)\mathrm{d}t}}$ KummerM(a, c, z)/GAMMA(c)=(GAMMA(b))/(2*Pi*I)*(z)^(1 - b)* int(exp(z*t)*(t)^(- b)* hypergeom([a , b], [c], (1)/(t)), t = - infinity..(0 + , 1 +)) Hypergeometric1F1Regularized[a, c, z]=Divide[Gamma[b],2*Pi*I]*(z)^(1 - b)* Integrate[Exp[z*t]*(t)^(- b)* HypergeometricPFQRegularized[{a , b}, {c}, Divide[1,t]], {t, - Infinity, (0 + , 1 +)}] Error Failure - Error
13.4.E13 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi% \mathrm{i}}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-% a}\mathrm{d}t}}$ KummerM(a, b, z)/GAMMA(b)=((z)^(1 - b))/(2*Pi*I)*int(exp(z*t)*(t)^(- b)*(1 -(1)/(t))^(- a), t = - infinity..(0 + , 1 +)) Hypergeometric1F1Regularized[a, b, z]=Divide[(z)^(1 - b),2*Pi*I]*Integrate[Exp[z*t]*(t)^(- b)*(1 -Divide[1,t])^(- a), {t, - Infinity, (0 + , 1 +)}] Error Failure - Error
13.4.E14 ${\displaystyle{\displaystyle U\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{% \Gamma\left(1-a\right)}{2\pi\mathrm{i}}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t% )^{b-a-1}}\mathrm{d}t}}$ KummerU(a, b, z)= exp(- a*Pi*I)*(GAMMA(1 - a))/(2*Pi*I)*int(exp(- z*t)*(t)^(a - 1)*(1 + t)^(b - a - 1), t = infinity..(0 +)) HypergeometricU[a, b, z]= Exp[- a*Pi*I]*Divide[Gamma[1 - a],2*Pi*I]*Integrate[Exp[- z*t]*(t)^(a - 1)*(1 + t)^(b - a - 1), {t, Infinity, (0 +)}] Error Failure - Error
13.4.E15 ${\displaystyle{\displaystyle\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)% \Gamma\left(c-b+1\right)}=\frac{z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e% ^{zt}t^{-c}{{}_{2}{\mathbf{F}}_{1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)% \mathrm{d}t}}$ (KummerU(a, b, z))/(GAMMA(c)*GAMMA(c - b + 1))=((z)^(1 - c))/(2*Pi*I)*int(exp(z*t)*(t)^(- c)* hypergeom([a , c], [a + c - b + 1], 1 -(1)/(t)), t = - infinity..(0 +)) Divide[HypergeometricU[a, b, z],Gamma[c]*Gamma[c - b + 1]]=Divide[(z)^(1 - c),2*Pi*I]*Integrate[Exp[z*t]*(t)^(- c)* HypergeometricPFQRegularized[{a , c}, {a + c - b + 1}, 1 -Divide[1,t]], {t, - Infinity, (0 +)}] Error Failure - Error
13.4.E16 ${\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi% \mathrm{i}\Gamma\left(a\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}% \frac{\Gamma\left(a+t\right)\Gamma\left(-t\right)}{\Gamma\left(b+t\right)}z^{t% }\mathrm{d}t}}$ KummerM(a, b, - z)/GAMMA(b)=(1)/(2*Pi*I*GAMMA(a))*int((GAMMA(a + t)*GAMMA(- t))/(GAMMA(b + t))*(z)^(t), t = - I*infinity..I*infinity) Hypergeometric1F1Regularized[a, b, - z]=Divide[1,2*Pi*I*Gamma[a]]*Integrate[Divide[Gamma[a + t]*Gamma[- t],Gamma[b + t]]*(z)^(t), {t, - I*Infinity, I*Infinity}] Failure Failure Skip Error
13.4.E17 ${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{-a}}{2\pi\mathrm{i}}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(1+a-b+t\right)\Gamma\left(-t\right)}{\Gamma\left(a\right)\Gamma\left(1+a% -b\right)}z^{-t}\mathrm{d}t}}$ KummerU(a, b, z)=((z)^(- a))/(2*Pi*I)*int((GAMMA(a + t)*GAMMA(1 + a - b + t)*GAMMA(- t))/(GAMMA(a)*GAMMA(1 + a - b))*(z)^(- t), t = - I*infinity..I*infinity) HypergeometricU[a, b, z]=Divide[(z)^(- a),2*Pi*I]*Integrate[Divide[Gamma[a + t]*Gamma[1 + a - b + t]*Gamma[- t],Gamma[a]*Gamma[1 + a - b]]*(z)^(- t), {t, - I*Infinity, I*Infinity}] Failure Failure Skip Error
13.4.E18 ${\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-b}e^{z}}{2\pi% \mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(b-1+t% \right)\Gamma\left(t\right)}{\Gamma\left(a+t\right)}z^{-t}\mathrm{d}t}}$ KummerU(a, b, z)=((z)^(1 - b)* exp(z))/(2*Pi*I)*int((GAMMA(b - 1 + t)*GAMMA(t))/(GAMMA(a + t))*(z)^(- t), t = - I*infinity..I*infinity) HypergeometricU[a, b, z]=Divide[(z)^(1 - b)* Exp[z],2*Pi*I]*Integrate[Divide[Gamma[b - 1 + t]*Gamma[t],Gamma[a + t]]*(z)^(- t), {t, - I*Infinity, I*Infinity}] Failure Failure Skip Error
13.6.E1 ${\displaystyle{\displaystyle M\left(a,a,z\right)=e^{z}}}$ KummerM(a, a, z)= exp(z) Hypergeometric1F1[a, a, z]= Exp[z] Successful Successful - -
13.6.E2 ${\displaystyle{\displaystyle M\left(1,2,2z\right)=\frac{e^{z}}{z}\sinh z}}$ KummerM(1, 2, 2*z)=(exp(z))/(z)*sinh(z) Hypergeometric1F1[1, 2, 2*z]=Divide[Exp[z],z]*Sinh[z] Successful Successful - -
13.6.E3 ${\displaystyle{\displaystyle M\left(0,b,z\right)=U\left(0,b,z\right)}}$ KummerM(0, b, z)= KummerU(0, b, z) Hypergeometric1F1[0, b, z]= HypergeometricU[0, b, z] Successful Successful - -
13.6.E3 ${\displaystyle{\displaystyle U\left(0,b,z\right)=1}}$ KummerU(0, b, z)= 1 HypergeometricU[0, b, z]= 1 Successful Successful - -
13.6.E4 ${\displaystyle{\displaystyle U\left(a,a+1,z\right)=z^{-a}}}$ KummerU(a, a + 1, z)= (z)^(- a) HypergeometricU[a, a + 1, z]= (z)^(- a) Failure Successful Successful -
13.6.E5