DLMF:14.19.E4 (Q4917): Difference between revisions

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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.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
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.m2acdec
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.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11593 / rank
 
Normal rank
Property / Symbols used: Q11593 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11593 / qualifier
 
xml-id: C14.S1.XMD5.m1dec
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.m1dec
Property / Symbols used
 
Property / Symbols used: Q11627 / rank
 
Normal rank
Property / Symbols used: Q11627 / qualifier
 
test:

ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}

\xi>0
Property / Symbols used: Q11627 / qualifier
 
xml-id: C14.S19.XMD8.m1dec

Latest revision as of 12:30, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.19.E4
No description defined

    Statements

    test
    P n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + m + 1 2 ) ⁢ ( sinh ⁡ ξ ) m 2 m ⁢ π 1 / 2 ⁢ Γ ⁡ ( n - m + 1 2 ) ⁢ Γ ⁡ ( m + 1 2 ) ⁢ ∫ 0 π ( sin ⁡ ϕ ) 2 ⁢ m ( cosh ⁡ ξ + cos ⁡ ϕ ⁢ sinh ⁡ ξ ) n + m + ( 1 / 2 ) ⁢ d ϕ , Legendre-P-first-kind 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 𝑚 1 2 superscript 𝜉 𝑚 superscript 2 𝑚 superscript 𝜋 1 2 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑚 1 2 superscript subscript 0 𝜋 superscript italic-ϕ 2 𝑚 superscript 𝜉 italic-ϕ 𝜉 𝑛 𝑚 1 2 italic-ϕ {\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{% \Gamma\left(n+m+\frac{1}{2}\right)(\sinh\xi)^{m}}{2^{m}\pi^{1/2}\Gamma\left(n-% m+\frac{1}{2}\right)\Gamma\left(m+\frac{1}{2}\right)}\*\int_{0}^{\pi}\frac{(% \sin\phi)^{2m}}{(\cosh\xi+\cos\phi\sinh\xi)^{n+m+(1/2)}}\mathrm{d}\phi,}}
    0 references
    DLMF id
    DLMF:14.19.E4
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    Q11481
    test
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2acdec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aadec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11593
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1dec
    0 references
    Q11602
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1dec
    0 references
    Q11627
    test
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD8.m1dec
    0 references
     
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