DLMF:11.4.E22 (Q3954)

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DLMF:11.4.E22
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    Statements

    Defining formula
    𝐇 1 ⁡ ( z ) = 2 π ⁢ ( 1 - J 0 ⁡ ( z ) ) + 4 π ⁢ ∑ k = 1 ∞ J 2 ⁢ k ⁡ ( z ) 4 ⁢ k 2 - 1 = 4 ⁢ ∑ k = 0 ∞ J 2 ⁢ k + 1 2 ⁡ ( 1 2 ⁢ z ) ⁢ J 2 ⁢ k + 3 2 ⁡ ( 1 2 ⁢ z ) . Struve-H 1 𝑧 2 𝜋 1 Bessel-J 0 𝑧 4 𝜋 superscript subscript 𝑘 1 Bessel-J 2 𝑘 𝑧 4 superscript 𝑘 2 1 4 superscript subscript 𝑘 0 Bessel-J 2 𝑘 1 2 1 2 𝑧 Bessel-J 2 𝑘 3 2 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{1}\left(z\right)=\frac{2}{\pi}(1-J_{0}% \left(z\right))+\frac{4}{\pi}\sum_{k=1}^{\infty}\frac{J_{2k}\left(z\right)}{4k% ^{2}-1}=4\sum_{k=0}^{\infty}J_{2k+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{2k+% \frac{3}{2}}\left(\tfrac{1}{2}z\right).}}
    0 references
    DLMF id
    DLMF:11.4.E22
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aedec
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    Struve function
    Defining formula
    𝐇 ν ⁡ ( z ) Struve-H 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S2.E1.m2aldec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2apdec
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    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1tdec
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    Q11497
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C11.S1.XMD5.m1ddec
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