DLMF:10.32.E1 (Q3521): Difference between revisions

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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: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1dec

Latest revision as of 13:40, 2 January 2020

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DLMF:10.32.E1
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    Statements

    Defining formula
    I 0 ⁡ ( z ) = 1 π ⁢ ∫ 0 π e ± z ⁢ cos ⁡ θ ⁢ d θ = 1 π ⁢ ∫ 0 π cosh ⁡ ( z ⁢ cos ⁡ θ ) ⁢ d θ . modified-Bessel-first-kind 0 𝑧 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 plus-or-minus 𝑧 𝜃 𝜃 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜃 {\displaystyle{\displaystyle I_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^% {\pm z\cos\theta}\mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cosh\left(z\cos% \theta\right)\mathrm{d}\theta.}}
    0 references
    DLMF id
    DLMF:10.32.E1
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    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.m1adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    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
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1dec
    0 references
     
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