DLMF:14.20.E17 (Q4938): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11635 / rank
 
Normal rank
Property / Symbols used: Q11635 / qualifier
 
Defining formula:

η 𝜂 {\displaystyle{\displaystyle\eta}}

\eta
Property / Symbols used: Q11635 / qualifier
 
xml-id: C14.S20.XMD7.m1dec

Latest revision as of 13:36, 2 January 2020

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DLMF:14.20.E17
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    Statements

    Defining formula
    𝖯 - 1 2 + i ⁢ τ - μ ⁡ ( x ) = σ ⁢ ( μ , τ ) ⁢ ( α 2 + η 1 + α 2 - x 2 ) 1 / 4 ⁢ I μ ⁡ ( τ ⁢ η 1 / 2 ) ⁢ ( 1 + O ⁡ ( 1 / τ ) ) , Ferrers-Legendre-P-first-kind 𝜇 1 2 𝑖 𝜏 𝑥 𝜎 𝜇 𝜏 superscript superscript 𝛼 2 𝜂 1 superscript 𝛼 2 superscript 𝑥 2 1 4 modified-Bessel-first-kind 𝜇 𝜏 superscript 𝜂 1 2 1 Big-O 1 𝜏 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x% \right)=\sigma(\mu,\tau)\left(\frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}}\right% )^{1/4}I_{\mu}\left(\tau\eta^{1/2}\right)\*\left(1+O\left(\ifrac{1}{\tau}% \right)\right),}}
    0 references
    DLMF id
    DLMF:14.20.E17
    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.m2abdec
    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.m2aidec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2awdec
    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.m2aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1tdec
    0 references
    Q11630
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C14.S1.XMD3.m1xdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1ldec
    0 references
    Q11633
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S20.XMD5.m1dec
    0 references
    Q11634
    Defining formula
    σ ⁢ ( μ , τ ) 𝜎 𝜇 𝜏 {\displaystyle{\displaystyle\sigma(\mu,\tau)}}
    xml-id
    C14.S20.XMD6.m1dec
    0 references
    Q11635
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C14.S20.XMD7.m1dec
    0 references
     
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