DLMF:10.11.E8 (Q3111)

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DLMF:10.11.E8
No description defined

    Statements

    Defining formula
    H n ( 2 ) ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = ( - 1 ) m ⁢ n ⁢ ( m ⁢ H n ( 1 ) ⁡ ( z ) + ( m + 1 ) ⁢ H n ( 2 ) ⁡ ( z ) ) . Hankel-H-2-Bessel-third-kind 𝑛 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 1 𝑚 𝑛 𝑚 Hankel-H-1-Bessel-third-kind 𝑛 𝑧 𝑚 1 Hankel-H-2-Bessel-third-kind 𝑛 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn}(m{% H^{(1)}_{n}}\left(z\right)+(m+1){H^{(2)}_{n}}\left(z\right)).}}
    0 references
    DLMF id
    DLMF:10.11.E8
    0 references
    Symbols used
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E5.m2addec
    0 references
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E6.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
    0 references
    Q11438
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C10.S1.XMD1.m1fdec
    0 references
    Q11429
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1bdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1gdec
    0 references
     
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