DLMF:11.7.E7 (Q3996): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11491 / rank
 
Normal rank
Property / Symbols used: Q11491 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11491 / qualifier
 
xml-id: C11.S1.XMD2.m1fdec

Revision as of 14:50, 2 January 2020

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DLMF:11.7.E7
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    Statements

    Defining formula
    ∫ 0 π / 2 𝐇 ν ⁡ ( z ⁢ sin ⁡ θ ) ⁢ ( sin ⁡ θ ) ν + 1 ( cos ⁡ θ ) 2 ⁢ ν ⁢ d θ = 2 - ν π ⁢ Γ ⁡ ( 1 2 - ν ) ⁢ z ν - 1 ⁢ ( 1 - cos ⁡ z ) , superscript subscript 0 𝜋 2 Struve-H 𝜈 𝑧 𝜃 superscript 𝜃 𝜈 1 superscript 𝜃 2 𝜈 𝜃 superscript 2 𝜈 𝜋 Euler-Gamma 1 2 𝜈 superscript 𝑧 𝜈 1 1 𝑧 {\displaystyle{\displaystyle\int_{0}^{\pi/2}\mathbf{H}_{\nu}\left(z\sin\theta% \right)\frac{(\sin\theta)^{\nu+1}}{(\cos\theta)^{2\nu}}\mathrm{d}\theta=\frac{% 2^{-\nu}}{\sqrt{\pi}}\Gamma\left(\tfrac{1}{2}-\nu\right)z^{\nu-1}(1-\cos z),}}
    0 references
    DLMF id
    DLMF:11.7.E7
    0 references
    constraint
    - 3 2 < ℜ ⁡ ν < 1 2 3 2 𝜈 1 2 {\displaystyle{\displaystyle-\tfrac{3}{2}<\Re\nu<\tfrac{1}{2}}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1fdec
    0 references
     
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