DLMF:10.7.E6 (Q3059): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosecant function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosecant function / qualifier
 
Defining formula:

csch ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}

\csch@@{\NVar{z}}
Property / Symbols used: hyperbolic cosecant function / qualifier
 
xml-id: C4.S28.E5.m2adec
Property / Symbols used
 
Property / Symbols used: hyperbolic cotangent function / rank
 
Normal rank
Property / Symbols used: hyperbolic cotangent function / qualifier
 
Defining formula:

coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}

\coth@@{\NVar{z}}
Property / Symbols used: hyperbolic cotangent function / qualifier
 
xml-id: C4.S28.E7.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: order less than / rank
 
Normal rank
Property / Symbols used: order less than / qualifier
 
Defining formula:

o ⁡ ( x ) little-o 𝑥 {\displaystyle{\displaystyle o\left(\NVar{x}\right)}}

\littleo@{\NVar{x}}
Property / Symbols used: order less than / qualifier
 
xml-id: C2.S1.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1adec
Property / Symbols used
 
Property / Symbols used: Q10812 / rank
 
Normal rank
Property / Symbols used: Q10812 / qualifier
 
Defining formula:

ℝ {\displaystyle{\displaystyle\mathbb{R}}}

\Reals
Property / Symbols used: Q10812 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r14.m2adec
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1edec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1cdec

Latest revision as of 15:09, 2 January 2020

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English
DLMF:10.7.E6
No description defined

    Statements

    Defining formula
    Y i ⁢ ν ⁡ ( z ) = i ⁢ csch ⁡ ( ν ⁢ π ) Γ ⁡ ( 1 - i ⁢ ν ) ⁢ ( 1 2 ⁢ z ) - i ⁢ ν - i ⁢ coth ⁡ ( ν ⁢ π ) Γ ⁡ ( 1 + i ⁢ ν ) ⁢ ( 1 2 ⁢ z ) i ⁢ ν + e | ν ⁢ ph ⁡ z | ⁢ o ⁡ ( 1 ) , Bessel-Y-Weber 𝑖 𝜈 𝑧 𝑖 𝜈 𝜋 Euler-Gamma 1 𝑖 𝜈 superscript 1 2 𝑧 𝑖 𝜈 𝑖 hyperbolic-cotangent 𝜈 𝜋 Euler-Gamma 1 𝑖 𝜈 superscript 1 2 𝑧 𝑖 𝜈 superscript 𝑒 𝜈 phase 𝑧 little-o 1 {\displaystyle{\displaystyle Y_{i\nu}\left(z\right)=\frac{i\operatorname{csch}% \left(\nu\pi\right)}{\Gamma\left(1-i\nu\right)}(\tfrac{1}{2}z)^{-i\nu}-\frac{i% \coth\left(\nu\pi\right)}{\Gamma\left(1+i\nu\right)}(\tfrac{1}{2}z)^{i\nu}+e^{% |\nu\operatorname{ph}z|}o\left(1\right),}}
    0 references
    DLMF id
    DLMF:10.7.E6
    0 references
    constraint
    ν ∈ ℝ 𝜈 {\displaystyle{\displaystyle\nu\in\mathbb{R}}}
    0 references
    ν ∈ ℝ 𝜈 ℝ {\displaystyle{\displaystyle\nu\in\mathbb{R}}}
    0 references
    ν ≠ 0 𝜈 0 {\displaystyle{\displaystyle\nu\neq 0}}
    0 references
    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    hyperbolic cosecant function
    Defining formula
    csch ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}
    xml-id
    C4.S28.E5.m2adec
    0 references
    hyperbolic cotangent function
    Defining formula
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    order less than
    Defining formula
    o ⁡ ( x ) little-o 𝑥 {\displaystyle{\displaystyle o\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E2.m2adec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
    Q10812
    Defining formula
    ℝ {\displaystyle{\displaystyle\mathbb{R}}}
    xml-id
    introduction.Sx4.p2.t1.r14.m2adec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1edec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1cdec
    0 references
     
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