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DLMF:13.2.E37 - MaRDI portal

DLMF:13.2.E37 (Q5026)

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DLMF:13.2.E37
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    Statements

    DLMF defining formula
    𝒲 ⁑ { z 1 - b ⁒ 𝐌 ⁑ ( a - b + 1 , 2 - b , z ) , e z ⁒ U ⁑ ( b - a , b , e Β± Ο€ ⁒ i ⁒ z ) } = - z - b ⁒ e z / Ξ“ ⁑ ( 1 - a ) , Wronskian superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 1 2 𝑏 𝑧 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-U 𝑏 π‘Ž 𝑏 superscript 𝑒 plus-or-minus πœ‹ imaginary-unit 𝑧 superscript 𝑧 𝑏 superscript 𝑒 𝑧 Euler-Gamma 1 π‘Ž {\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^{\pm\pi\mathrm{i}}z\right)\right\}=-\ifrac{z^{% -b}e^{z}}{\Gamma\left(1-a\right)},}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2apdec
    0 references
    Kummer confluent hypergeometric function
    DLMF defining formula
    U ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2audec
    0 references
    Olver’s confluent hypergeometric function
    DLMF defining formula
    𝐌 ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2aidec
    0 references
    Wronskian
    DLMF defining formula
    𝒲 Wronskian {\displaystyle{\displaystyle\mathscr{W}}}
    xml-id
    C1.S13.Px2.p1.m3addec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ajdec
    0 references

    Identifiers

     
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