DLMF:12.13.E5 (Q4234): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11516 / rank
 
Normal rank
Property / Symbols used: Q11516 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11516 / qualifier
 
xml-id: C12.S1.XMD6.m1ddec

Latest revision as of 15:26, 2 January 2020

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DLMF:12.13.E5
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    Statements

    Defining formula
    U ⁡ ( a , x ⁢ cos ⁡ t + y ⁢ sin ⁡ t ) = e 1 4 ⁢ ( x ⁢ sin ⁡ t - y ⁢ cos ⁡ t ) 2 ⁢ ∑ m = 0 ∞ ( - a - 1 2 m ) ⁢ ( tan ⁡ t ) m ⁢ U ⁡ ( m + a , x ) ⁢ U ⁡ ( - m - 1 2 , y ) , parabolic-U 𝑎 𝑥 𝑡 𝑦 𝑡 superscript 𝑒 1 4 superscript 𝑥 𝑡 𝑦 𝑡 2 superscript subscript 𝑚 0 binomial 𝑎 1 2 𝑚 superscript 𝑡 𝑚 parabolic-U 𝑚 𝑎 𝑥 parabolic-U 𝑚 1 2 𝑦 {\displaystyle{\displaystyle U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right),}}
    0 references
    DLMF id
    DLMF:12.13.E5
    0 references
    constraint
    ℜ ⁡ a ≤ - 1 2 , 0 ≤ t ≤ 1 4 ⁢ π formulae-sequence 𝑎 1 2 0 𝑡 1 4 𝜋 {\displaystyle{\displaystyle\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi}}
    0 references
    Symbols used
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    tangent function
    Defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2adec
    0 references
    Q11520
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C12.S1.XMD1.m1ddec
    0 references
    Q11542
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C12.S1.XMD2.m1ddec
    0 references
    Q11516
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C12.S1.XMD6.m1ddec
    0 references
     
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