DLMF:13.2.E12 (Q4302)

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DLMF:13.2.E12
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    Statements

    Defining formula
    U ⁡ ( a , b , z ⁢ e 2 ⁢ π ⁢ i ⁢ m ) = 2 ⁢ π ⁢ i ⁢ e - π ⁢ i ⁢ b ⁢ m ⁢ sin ⁡ ( π ⁢ b ⁢ m ) Γ ⁡ ( 1 + a - b ) ⁢ sin ⁡ ( π ⁢ b ) ⁢ 𝐌 ⁡ ( a , b , z ) + e - 2 ⁢ π ⁢ i ⁢ b ⁢ m ⁢ U ⁡ ( a , b , z ) . Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑒 2 𝜋 imaginary-unit 𝑚 2 𝜋 imaginary-unit superscript 𝑒 𝜋 imaginary-unit 𝑏 𝑚 𝜋 𝑏 𝑚 Euler-Gamma 1 𝑎 𝑏 𝜋 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑒 2 𝜋 imaginary-unit 𝑏 𝑚 Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\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).}}
    0 references
    DLMF id
    DLMF:13.2.E12
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2afdec
    0 references
    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
     
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