DLMF:13.4.E13 (Q4376)

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DLMF:13.4.E13
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    Statements

    Defining formula
    𝐌 ⁑ ( a , b , z ) = z 1 - b 2 ⁒ Ο€ ⁒ i ⁒ ∫ - ∞ ( 0 + , 1 + ) e z ⁒ t ⁒ t - b ⁒ ( 1 - 1 t ) - a ⁒ d t , Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 superscript 𝑧 1 𝑏 2 πœ‹ imaginary-unit superscript subscript limit-from 0 limit-from 1 superscript 𝑒 𝑧 𝑑 superscript 𝑑 𝑏 superscript 1 1 𝑑 π‘Ž 𝑑 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi% \mathrm{i}}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-% a}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E13
    0 references
    constraint
    | ph ⁑ z | < 1 2 ⁒ Ο€ phase 𝑧 1 2 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\frac{1}{2}\pi}}
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    Symbols used
    Olver’s confluent hypergeometric function
    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.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
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    Q10777
    Defining formula
    ( a , b ) π‘Ž 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6aadec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aedec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ldec
    0 references
     
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