DLMF:30.6.E5 (Q8864): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12423 / rank
 
Normal rank
Property / Symbols used: Q12423 / qualifier
 
Defining formula:

𝑄𝑠 n m ⁡ ( z , γ 2 ) spheroidal-wave-Qs 𝑚 𝑛 𝑧 superscript 𝛾 2 {\displaystyle{\displaystyle\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},% \NVar{\gamma^{2}}\right)}}

\sphwaveQs{\NVar{m}}{\NVar{n}}@{\NVar{z}}{\NVar{\gamma^{2}}}
Property / Symbols used: Q12423 / qualifier
 
xml-id: C30.S6.p1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q12418 / rank
 
Normal rank
Property / Symbols used: Q12418 / qualifier
 
Defining formula:

𝖯𝗌 n m ⁡ ( x , γ 2 ) spheroidal-wave-on-cut-Ps 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},% \NVar{\gamma^{2}}\right)}}

\sphwavePsreal{\NVar{m}}{\NVar{n}}@{\NVar{x}}{\NVar{\gamma^{2}}}
Property / Symbols used: Q12418 / qualifier
 
xml-id: C30.S4.SS1.m1aadec
Property / Symbols used
 
Property / Symbols used: Q12421 / rank
 
Normal rank
Property / Symbols used: Q12421 / qualifier
 
Defining formula:

𝖰𝗌 n m ⁡ ( x , γ 2 ) spheroidal-wave-on-cut-Qs 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},% \NVar{\gamma^{2}}\right)}}

\sphwaveQsreal{\NVar{m}}{\NVar{n}}@{\NVar{x}}{\NVar{\gamma^{2}}}
Property / Symbols used: Q12421 / qualifier
 
xml-id: C30.S5.m1adec
Property / Symbols used
 
Property / Symbols used: Q12419 / rank
 
Normal rank
Property / Symbols used: Q12419 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12419 / qualifier
 
xml-id: C30.S1.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q12408 / rank
 
Normal rank
Property / Symbols used: Q12408 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12408 / qualifier
 
xml-id: C30.S1.XMD4.m1ddec
Property / Symbols used
 
Property / Symbols used: Q12409 / rank
 
Normal rank
Property / Symbols used: Q12409 / qualifier
 
Defining formula:

n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}

n\geq m
Property / Symbols used: Q12409 / qualifier
 
xml-id: C30.S1.XMD5.m1ddec
Property / Symbols used
 
Property / Symbols used: Q12404 / rank
 
Normal rank
Property / Symbols used: Q12404 / qualifier
 
Defining formula:

γ 2 superscript 𝛾 2 {\displaystyle{\displaystyle\gamma^{2}}}

\gamma^{2}
Property / Symbols used: Q12404 / qualifier
 
xml-id: C30.S2.XMD3.m1cdec

Latest revision as of 01:45, 2 January 2020

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English
DLMF:30.6.E5
No description defined

    Statements

    Defining formula
    𝑄𝑠 n m ⁡ ( x ± i ⁢ 0 , γ 2 ) = ( ∓ i ) m ⁢ ( 𝖰𝗌 n m ⁡ ( x , γ 2 ) ∓ 1 2 ⁢ i ⁢ π ⁢ 𝖯𝗌 n m ⁡ ( x , γ 2 ) ) . spheroidal-wave-Qs 𝑚 𝑛 plus-or-minus 𝑥 imaginary-unit 0 superscript 𝛾 2 superscript minus-or-plus imaginary-unit 𝑚 minus-or-plus spheroidal-wave-on-cut-Qs 𝑚 𝑛 𝑥 superscript 𝛾 2 1 2 imaginary-unit 𝜋 spheroidal-wave-on-cut-Ps 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathit{Qs}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2% }\right)={(\mp\mathrm{i})^{m}\left(\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right% )\mp\tfrac{1}{2}\mathrm{i}\pi\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)% \right)}.}}
    0 references
    DLMF id
    DLMF:30.6.E5
    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
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q12423
    Defining formula
    𝑄𝑠 n m ⁡ ( z , γ 2 ) spheroidal-wave-Qs 𝑚 𝑛 𝑧 superscript 𝛾 2 {\displaystyle{\displaystyle\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S6.p1.m2acdec
    0 references
    Q12418
    Defining formula
    𝖯𝗌 n m ⁡ ( x , γ 2 ) spheroidal-wave-on-cut-Ps 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S4.SS1.m1aadec
    0 references
    Q12421
    Defining formula
    𝖰𝗌 n m ⁡ ( x , γ 2 ) spheroidal-wave-on-cut-Qs 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S5.m1adec
    0 references
    Q12419
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C30.S1.XMD2.m1adec
    0 references
    Q12408
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C30.S1.XMD4.m1ddec
    0 references
    Q12409
    Defining formula
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
    xml-id
    C30.S1.XMD5.m1ddec
    0 references
    Q12404
    Defining formula
    γ 2 superscript 𝛾 2 {\displaystyle{\displaystyle\gamma^{2}}}
    xml-id
    C30.S2.XMD3.m1cdec
    0 references
     
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