DLMF:14.20.E15 (Q4936): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Ferrers function of the first kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the first kind / qualifier
 
Defining formula:

𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}

\FerrersP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the first kind / qualifier
 
xml-id: C14.S3.E1.m2ahdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

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

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2aedec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2audec
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
Defining formula:

I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselI{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
xml-id: C10.S25.E2.m2adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

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

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11630 / rank
 
Normal rank
Property / Symbols used: Q11630 / qualifier
 
Defining formula:

τ 𝜏 {\displaystyle{\displaystyle\tau}}

\tau
Property / Symbols used: Q11630 / qualifier
 
xml-id: C14.S1.XMD3.m1vdec
Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11632 / rank
 
Normal rank
Property / Symbols used: Q11632 / qualifier
 
Defining formula:

0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}

0<\theta<\pi
Property / Symbols used: Q11632 / qualifier
 
xml-id: C14.S20.XMD2.m1dec

Latest revision as of 13:35, 2 January 2020

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DLMF:14.20.E15
No description defined

    Statements

    Defining formula
    𝖯 - 1 2 + i ⁢ τ - μ ⁡ ( cos ⁡ θ ) = 1 τ μ ⁢ ( θ sin ⁡ θ ) 1 / 2 ⁢ I μ ⁡ ( τ ⁢ θ ) ⁢ ( 1 + O ⁡ ( 1 / τ ) ) , Ferrers-Legendre-P-first-kind 𝜇 1 2 𝑖 𝜏 𝜃 1 superscript 𝜏 𝜇 superscript 𝜃 𝜃 1 2 modified-Bessel-first-kind 𝜇 𝜏 𝜃 1 Big-O 1 𝜏 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\cos% \theta\right)=\frac{1}{\tau^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}% I_{\mu}\left(\tau\theta\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),}}
    0 references
    DLMF id
    DLMF:14.20.E15
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2ahdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2audec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
    Q11630
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C14.S1.XMD3.m1vdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1jdec
    0 references
    Q11632
    Defining formula
    0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}
    xml-id
    C14.S20.XMD2.m1dec
    0 references
     
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