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DLMF:10.41.E5 - MaRDI portal

DLMF:10.41.E5 (Q4294)

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DLMF:10.41.E5
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    Statements

    DLMF defining formula
    I ν ′ ⁡ ( ν ⁢ z ) ∼ ( 1 + z 2 ) 1 4 ⁢ e ν ⁢ η ( 2 ⁢ π ⁢ ν ) 1 2 ⁢ z ⁢ ∑ k = 0 ∞ V k ⁢ ( p ) ν k , asymptotic-expansion diffop modified-Bessel-first-kind 𝜈 1 𝜈 𝑧 superscript 1 superscript 𝑧 2 1 4 superscript 𝑒 𝜈 𝜂 superscript 2 𝜋 𝜈 1 2 𝑧 superscript subscript 𝑘 0 subscript 𝑉 𝑘 𝑝 superscript 𝜈 𝑘 {\displaystyle{\displaystyle I_{\nu}'\left(\nu z\right)\sim\frac{(1+z^{2})^{% \frac{1}{4}}e^{\nu\eta}}{(2\pi\nu)^{\frac{1}{2}}z}\sum_{k=0}^{\infty}\frac{V_{% k}(p)}{\nu^{k}},}}
    0 references
    Symbols used
    Q10905
    DLMF defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    modified Bessel function of the first kind
    DLMF defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2abdec
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    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1bdec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1ddec
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    complex parameter
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ddec
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    10.41.E7
    DLMF defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C10.S41.XMD1.m1bdec
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    polynomial coefficient
    DLMF defining formula
    V k ⁢ ( p ) subscript 𝑉 𝑘 𝑝 {\displaystyle{\displaystyle V_{k}(p)}}
    xml-id
    C10.S41.XMD4.m1dec
    0 references

    Identifiers

     
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