DLMF:10.56.E5 (Q3769)

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DLMF:10.56.E5
No description defined

    Statements

    Defining formula
    exp ⁡ ( - z 2 + 2 ⁢ i ⁢ z ⁢ t ) z = e - z z + 2 π ⁢ ∑ n = 1 ∞ ( - i ⁢ t ) n n ! ⁢ 𝗄 n - 1 ⁡ ( z ) . superscript 𝑧 2 2 𝑖 𝑧 𝑡 𝑧 superscript 𝑒 𝑧 𝑧 2 𝜋 superscript subscript 𝑛 1 superscript 𝑖 𝑡 𝑛 𝑛 spherical-Bessel-K 𝑛 1 𝑧 {\displaystyle{\displaystyle\frac{\exp\left(-\sqrt{z^{2}+2izt}\right)}{z}=% \frac{e^{-z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\mathsf{k% }_{n-1}\left(z\right).}}
    0 references
    DLMF id
    DLMF:10.56.E5
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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    modified spherical Bessel function
    Defining formula
    𝗄 n ⁡ ( z ) spherical-Bessel-K 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E9.m2adec
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    Q11429
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1ddec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1ddec
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