DLMF:10.45.E7 (Q3664): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
Defining formula:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: modified Bessel function fo the second kind of imaginary order / rank
 
Normal rank
Property / Symbols used: modified Bessel function fo the second kind of imaginary order / qualifier
 
Defining formula:

K ~ ν ⁡ ( x ) modified-Bessel-second-kind-imaginary-order 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)}}

\modBesselKimag{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: modified Bessel function fo the second kind of imaginary order / qualifier
 
xml-id: C10.S45.E2.m3addec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q11435 / rank
 
Normal rank
Property / Symbols used: Q11435 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11435 / qualifier
 
xml-id: C10.S1.XMD4.m1fdec
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.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11477 / rank
 
Normal rank
Property / Symbols used: Q11477 / qualifier
 
Defining formula:

γ ν subscript 𝛾 𝜈 {\displaystyle{\displaystyle\gamma_{\nu}}}

\gamma_{\nu}
Property / Symbols used: Q11477 / qualifier
 
xml-id: C10.S24.XMD1.m1adec

Latest revision as of 14:07, 2 January 2020

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DLMF:10.45.E7
No description defined

    Statements

    Defining formula
    K ~ ν ⁡ ( x ) = - ( π ν ⁢ sinh ⁡ ( π ⁢ ν ) ) 1 2 ⁢ sin ⁡ ( ν ⁢ ln ⁡ ( 1 2 ⁢ x ) - γ ν ) + O ⁡ ( x 2 ) , modified-Bessel-second-kind-imaginary-order 𝜈 𝑥 superscript 𝜋 𝜈 𝜋 𝜈 1 2 𝜈 1 2 𝑥 subscript 𝛾 𝜈 Big-O superscript 𝑥 2 {\displaystyle{\displaystyle\widetilde{K}_{\nu}\left(x\right)=-\left(\frac{\pi% }{\nu\sinh\left(\pi\nu\right)}\right)^{\frac{1}{2}}\*\sin\left(\nu\ln\left(% \tfrac{1}{2}x\right)-\gamma_{\nu}\right)+O\left(x^{2}\right),}}
    0 references
    DLMF id
    DLMF:10.45.E7
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aadec
    0 references
    modified Bessel function fo the second kind of imaginary order
    Defining formula
    K ~ ν ⁡ ( x ) modified-Bessel-second-kind-imaginary-order 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)}}
    xml-id
    C10.S45.E2.m3addec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1fdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1fdec
    0 references
    Q11477
    Defining formula
    γ ν subscript 𝛾 𝜈 {\displaystyle{\displaystyle\gamma_{\nu}}}
    xml-id
    C10.S24.XMD1.m1adec
    0 references
     
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