DLMF:10.43.E7 (Q3621)

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DLMF:10.43.E7
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    Statements

    Defining formula
    ∫ 0 x e ± t ⁢ t ν ⁢ I ν ⁡ ( t ) ⁢ d t = e ± x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( I ν ⁡ ( x ) ∓ I ν + 1 ⁡ ( x ) ) , superscript subscript 0 𝑥 superscript 𝑒 plus-or-minus 𝑡 superscript 𝑡 𝜈 modified-Bessel-first-kind 𝜈 𝑡 𝑡 superscript 𝑒 plus-or-minus 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 minus-or-plus modified-Bessel-first-kind 𝜈 𝑥 modified-Bessel-first-kind 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{\pm t}t^{\nu}I_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{\pm x}x^{\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)\mp I_{\nu+% 1}\left(x\right)),}}
    0 references
    DLMF id
    DLMF:10.43.E7
    0 references
    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    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.m2abdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
     
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