DLMF:14.19.E6 (Q4919): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
test:

cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2aedec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
test:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11602 / rank
 
Normal rank
Property / Symbols used: Q11602 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11602 / qualifier
 
xml-id: C14.S1.XMD6.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
test:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1bdec

Latest revision as of 12:30, 2 January 2020

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DLMF:14.19.E6
No description defined

    Statements

    test
    𝑸 - 1 2 μ ⁡ ( cosh ⁡ ξ ) + 2 ⁢ ∑ n = 1 ∞ Γ ⁡ ( μ + n + 1 2 ) Γ ⁡ ( μ + 1 2 ) ⁢ 𝑸 n - 1 2 μ ⁡ ( cosh ⁡ ξ ) ⁢ cos ⁡ ( n ⁢ ϕ ) = ( 1 2 ⁢ π ) 1 / 2 ⁢ ( sinh ⁡ ξ ) μ ( cosh ⁡ ξ - cos ⁡ ϕ ) μ + ( 1 / 2 ) , associated-Legendre-black-Q 𝜇 1 2 𝜉 2 superscript subscript 𝑛 1 Euler-Gamma 𝜇 𝑛 1 2 Euler-Gamma 𝜇 1 2 associated-Legendre-black-Q 𝜇 𝑛 1 2 𝜉 𝑛 italic-ϕ superscript 1 2 𝜋 1 2 superscript 𝜉 𝜇 superscript 𝜉 italic-ϕ 𝜇 1 2 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\left(\cosh\xi% \right)+2\sum_{n=1}^{\infty}\frac{\Gamma\left(\mu+n+\tfrac{1}{2}\right)}{% \Gamma\left(\mu+\tfrac{1}{2}\right)}\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\left(% \cosh\xi\right)\cos\left(n\phi\right)=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}% \left(\sinh\xi\right)^{\mu}}{\left(\cosh\xi-\cos\phi\right)^{\mu+(1/2)}},}}
    0 references
    DLMF id
    DLMF:14.19.E6
    0 references
    constraint
    ℜ ⁡ μ > - 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Q11592
    test
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aedec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2acdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11602
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1bdec
    0 references
    Q11591
    test
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1bdec
    0 references
     
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