DLMF:2.11.E15 (Q1040): 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.m2agdec
Property / Symbols used
 
Property / Symbols used: complementary error function / rank
 
Normal rank
Property / Symbols used: complementary error function / qualifier
 
Defining formula:

erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}

\erfc@@{\NVar{z}}
Property / Symbols used: complementary error function / qualifier
 
xml-id: C7.S2.E2.m2adec
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.m2ahdec
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.m2addec
Property / Symbols used
 
Property / Symbols used: terminant function / rank
 
Normal rank
Property / Symbols used: terminant function / qualifier
 
Defining formula:

F p ⁡ ( z ) terminant-function 𝑝 𝑧 {\displaystyle{\displaystyle F_{\NVar{p}}\left(\NVar{z}\right)}}

\terminant{\NVar{p}}@{\NVar{z}}
Property / Symbols used: terminant function / qualifier
 
xml-id: C2.S11.E11.m2addec
Property / Symbols used
 
Property / Symbols used: Q11136 / rank
 
Normal rank
Property / Symbols used: Q11136 / qualifier
 
Defining formula:

ρ 𝜌 {\displaystyle{\displaystyle\rho}}

\rho
Property / Symbols used: Q11136 / qualifier
 
xml-id: C2.S11.XMD3.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11137 / rank
 
Normal rank
Property / Symbols used: Q11137 / qualifier
 
Defining formula:

θ 𝜃 {\displaystyle{\displaystyle\theta}}

\theta
Property / Symbols used: Q11137 / qualifier
 
xml-id: C2.S11.XMD4.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11139 / rank
 
Normal rank
Property / Symbols used: Q11139 / qualifier
 
Defining formula:

c ⁢ ( θ ) 𝑐 𝜃 {\displaystyle{\displaystyle c(\theta)}}

c(\theta)
Property / Symbols used: Q11139 / qualifier
 
xml-id: C2.S11.XMD7.m1dec
Property / Symbols used
 
Property / Symbols used: Q11140 / rank
 
Normal rank
Property / Symbols used: Q11140 / qualifier
 
Defining formula:

h 2 ⁢ s ⁢ ( θ , α ) subscript ℎ 2 𝑠 𝜃 𝛼 {\displaystyle{\displaystyle h_{2s}(\theta,\alpha)}}

h_{2s}(\theta,\alpha)
Property / Symbols used: Q11140 / qualifier
 
xml-id: C2.S11.XMD8.m1dec

Latest revision as of 17:40, 1 January 2020

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English
DLMF:2.11.E15
No description defined

    Statements

    Defining formula
    F n + p ⁡ ( z ) ∼ ( - 1 ) n ⁢ i ⁢ e - p ⁢ π ⁢ i ⁢ ( 1 2 ⁢ erfc ⁡ ( 1 2 ⁢ ρ ⁢ c ⁢ ( θ ) ) - i ⁢ e i ⁢ ρ ⁢ ( π - θ ) ⁢ e - ρ - z ( 2 ⁢ π ⁢ ρ ) 1 / 2 ⁢ ∑ s = 0 ∞ h 2 ⁢ s ⁢ ( θ , α ) ρ s ) . asymptotic-expansion terminant-function 𝑛 𝑝 𝑧 superscript 1 𝑛 𝑖 superscript 𝑒 𝑝 𝜋 𝑖 1 2 complementary-error-function 1 2 𝜌 𝑐 𝜃 𝑖 superscript 𝑒 𝑖 𝜌 𝜋 𝜃 superscript 𝑒 𝜌 𝑧 superscript 2 𝜋 𝜌 1 2 superscript subscript 𝑠 0 subscript ℎ 2 𝑠 𝜃 𝛼 superscript 𝜌 𝑠 {\displaystyle{\displaystyle F_{n+p}\left(z\right)\sim(-1)^{n}ie^{-p\pi i}% \left(\tfrac{1}{2}\operatorname{erfc}\left(\sqrt{\tfrac{1}{2}\rho}\,c(\theta)% \right)-i\frac{e^{i\rho(\pi-\theta)}e^{-\rho-z}}{(2\pi\rho)^{1/2}}\sum_{s=0}^{% \infty}\frac{h_{2s}(\theta,\alpha)}{\rho^{s}}\right).}}
    0 references
    DLMF id
    DLMF:2.11.E15
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aedec
    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
    0 references
    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    terminant function
    Defining formula
    F p ⁡ ( z ) terminant-function 𝑝 𝑧 {\displaystyle{\displaystyle F_{\NVar{p}}\left(\NVar{z}\right)}}
    xml-id
    C2.S11.E11.m2addec
    0 references
    Q11136
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C2.S11.XMD3.m1cdec
    0 references
    Q11137
    Defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C2.S11.XMD4.m1cdec
    0 references
    Q11139
    Defining formula
    c ⁢ ( θ ) 𝑐 𝜃 {\displaystyle{\displaystyle c(\theta)}}
    xml-id
    C2.S11.XMD7.m1dec
    0 references
    Q11140
    Defining formula
    h 2 ⁢ s ⁢ ( θ , α ) subscript ℎ 2 𝑠 𝜃 𝛼 {\displaystyle{\displaystyle h_{2s}(\theta,\alpha)}}
    xml-id
    C2.S11.XMD8.m1dec
    0 references
     
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