DLMF:10.34.E3 (Q3544)

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DLMF:10.34.E3
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    Statements

    test
    I ν ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( i / π ) ⁢ ( ± e m ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ⁢ e ± π ⁢ i ) ∓ e ( m ∓ 1 ) ⁢ ν ⁢ π ⁢ i ⁢ K ν ⁡ ( z ) ) , modified-Bessel-first-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 𝑖 𝜋 minus-or-plus plus-or-minus superscript 𝑒 𝑚 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 superscript 𝑒 minus-or-plus 𝑚 1 𝜈 𝜋 𝑖 modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\nu}\left(ze^{m\pi i}\right)=(i/\pi)\left(\pm e% ^{m\nu\pi i}K_{\nu}\left(ze^{\pm\pi i}\right)\mp e^{(m\mp 1)\nu\pi i}K_{\nu}% \left(z\right)\right),}}
    0 references
    DLMF id
    DLMF:10.34.E3
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    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.m2abdec
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    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2aadec
    0 references
    Q11438
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C10.S1.XMD1.m1bdec
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    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1bdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1bdec
    0 references
     
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