DLMF:11.6.E3 (Q4676)

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DLMF:11.6.E3
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    Statements

    DLMF defining formula
    ∫ 0 z 𝐊 0 ⁑ ( t ) ⁒ d t - 2 Ο€ ⁒ ( ln ⁑ ( 2 ⁒ z ) + Ξ³ ) ∼ 2 Ο€ ⁒ βˆ‘ k = 1 ∞ ( - 1 ) k + 1 ⁒ ( 2 ⁒ k ) ! ⁒ ( 2 ⁒ k - 1 ) ! ( k ! ) 2 ⁒ ( 2 ⁒ z ) 2 ⁒ k , asymptotic-expansion superscript subscript 0 𝑧 associated-Struve-K 0 𝑑 𝑑 2 πœ‹ 2 𝑧 2 πœ‹ superscript subscript π‘˜ 1 superscript 1 π‘˜ 1 2 π‘˜ 2 π‘˜ 1 superscript π‘˜ 2 superscript 2 𝑧 2 π‘˜ {\displaystyle{\displaystyle\int_{0}^{z}\mathbf{K}_{0}\left(t\right)\mathrm{d}% t-\frac{2}{\pi}(\ln\left(2z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}% (-1)^{k+1}\frac{(2k)!(2k-1)!}{(k!)^{2}(2z)^{2k}},}}
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    constraint
    | ph ⁑ z | ≀ Ο€ - Ξ΄ phase 𝑧 πœ‹ 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\pi-\delta}}
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    Symbols used
    Euler’s constant
    DLMF defining formula
    Ξ³ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
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    Struve function
    DLMF defining formula
    𝐊 Ξ½ ⁑ ( z ) associated-Struve-K 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S2.E5.m2aadec
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    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
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    differential of x
    DLMF defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    principal branch of logarithm function
    DLMF defining formula
    ln ⁑ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
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    phase
    DLMF defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1bdec
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    nonnegative integer
    DLMF defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C11.S1.XMD5.m1bdec
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    arbitrary small positive constant
    DLMF defining formula
    Ξ΄ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C11.S1.XMD6.m1bdec
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    Identifiers

     
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