DLMF:2.5.E10 (Q786)

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DLMF:2.5.E10
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    Statements

    Defining formula
    ℳ ⁡ h ⁡ ( z ) = 2 z - 1 ⁢ Γ ⁡ ( ν + 1 2 ⁢ z ) Γ 2 ⁡ ( 1 - 1 2 ⁢ z ) ⁢ Γ ⁡ ( 1 + ν - 1 2 ⁢ z ) ⁢ Γ ⁡ ( z ) ⁢ π sin ⁡ ( π ⁢ z ) , Mellin-transform ℎ 𝑧 superscript 2 𝑧 1 Euler-Gamma 𝜈 1 2 𝑧 Euler-Gamma 2 1 1 2 𝑧 Euler-Gamma 1 𝜈 1 2 𝑧 Euler-Gamma 𝑧 𝜋 𝜋 𝑧 {\displaystyle{\displaystyle\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z% \right)=\frac{2^{z-1}\Gamma\left(\nu+\frac{1}{2}z\right)}{{\Gamma^{2}}\left(1-% \frac{1}{2}z\right)\Gamma\left(1+\nu-\frac{1}{2}z\right)\Gamma\left(z\right)}% \frac{\pi}{\sin\left(\pi z\right)},}}
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    DLMF id
    DLMF:2.5.E10
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    constraint
    - 2 ⁢ ν < ℜ ⁡ z < 1 2 𝜈 𝑧 1 {\displaystyle{\displaystyle-2\nu<\Re z<1}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    Mellin transform
    Defining formula
    ℳ ⁡ ( f ) ⁡ ( s ) Mellin-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E32.m2agdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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    Q10999
    Defining formula
    h ⁢ ( t ) = J ν 2 ⁡ ( t ) ℎ 𝑡 Bessel-J 𝜈 2 𝑡 {\displaystyle{\displaystyle h(t)={J_{\nu}^{2}}\left(t\right)}}
    xml-id
    C2.S5.XMD10.m1dec
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    Q10998
    Defining formula
    f ⁢ ( t ) = 1 / ( 1 + t ) 𝑓 𝑡 1 1 𝑡 {\displaystyle{\displaystyle f(t)=1/(1+t)}}
    xml-id
    C2.S5.XMD9.m1adec
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