DLMF:10.38.E1 (Q3560): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2adec

Revision as of 13:48, 2 January 2020

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DLMF:10.38.E1
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    Statements

    test
    ∂ ⁡ I ± ν ⁡ ( z ) ∂ ⁡ ν = ± I ± ν ⁡ ( z ) ⁢ ln ⁡ ( 1 2 ⁢ z ) ∓ ( 1 2 ⁢ z ) ± ν ⁢ ∑ k = 0 ∞ ψ ⁡ ( k + 1 ± ν ) Γ ⁡ ( k + 1 ± ν ) ⁢ ( 1 4 ⁢ z 2 ) k k ! , partial-derivative modified-Bessel-first-kind plus-or-minus 𝜈 𝑧 𝜈 minus-or-plus plus-or-minus modified-Bessel-first-kind plus-or-minus 𝜈 𝑧 1 2 𝑧 superscript 1 2 𝑧 plus-or-minus 𝜈 superscript subscript 𝑘 0 digamma plus-or-minus 𝑘 1 𝜈 Euler-Gamma plus-or-minus 𝑘 1 𝜈 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 {\displaystyle{\displaystyle\frac{\partial I_{\pm\nu}\left(z\right)}{\partial% \nu}=\pm I_{\pm\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}% z)^{\pm\nu}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+% 1\pm\nu\right)}\frac{(\frac{1}{4}z^{2})^{k}}{k!},}}
    0 references
    DLMF id
    DLMF:10.38.E1
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    modified Bessel function of the first kind
    test
    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
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
     
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