DLMF:12.12.E4 (Q4926)

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DLMF:12.12.E4
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    Statements

    DLMF defining formula
    ( U ⁡ ( a , z ) ) 2 + ( U ¯ ⁡ ( a , z ) ) 2 = 2 3 2 π ⁢ Γ ⁡ ( 1 2 - a ) ⁢ ∫ 0 ∞ e 2 ⁢ a ⁢ t + 1 2 ⁢ z 2 ⁢ tanh ⁡ t sinh ⁡ ( 2 ⁢ t ) ⁢ d t , superscript parabolic-U 𝑎 𝑧 2 superscript parabolic-U-bar 𝑎 𝑧 2 superscript 2 3 2 𝜋 Euler-Gamma 1 2 𝑎 superscript subscript 0 superscript 𝑒 2 𝑎 𝑡 1 2 superscript 𝑧 2 𝑡 2 𝑡 𝑡 {\displaystyle{\displaystyle(U\left(a,z\right))^{2}+(\overline{U}\left(a,z% \right))^{2}=\frac{2^{\frac{3}{2}}}{\pi}\Gamma\left(\tfrac{1}{2}-a\right)\int_% {0}^{\infty}\frac{e^{2at+\frac{1}{2}z^{2}\tanh t}}{\sqrt{\sinh\left(2t\right)}% }\mathrm{d}t,}}
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    constraint
    ℜ ⁡ a < 1 2 𝑎 1 2 {\displaystyle{\displaystyle\Re a<\tfrac{1}{2}}}
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    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
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    hyperbolic sine function
    DLMF defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
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    hyperbolic tangent function
    DLMF defining formula
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2adec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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    Q11291
    DLMF defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1acdec
    0 references
    Q11490
    DLMF defining formula
    U ¯ ⁡ ( a , x ) parabolic-U-bar 𝑎 𝑥 {\displaystyle{\displaystyle\overline{U}\left(\NVar{a},\NVar{x}\right)}}
    xml-id
    C12.S2.SS6.p1.m4adec
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    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1dec
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    real or complex parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C12.S1.XMD6.m1cdec
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    Identifiers

     
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