DLMF:8.12.E16 (Q2618): 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
 
test:

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

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2addec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

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: Q11326 / rank
 
Normal rank
Property / Symbols used: Q11326 / qualifier
 
test:

Q ⁡ ( a , z ) incomplete-gamma-Q 𝑎 𝑧 {\displaystyle{\displaystyle Q\left(\NVar{a},\NVar{z}\right)}}

\normincGammaQ@{\NVar{a}}{\NVar{z}}
Property / Symbols used: Q11326 / qualifier
 
xml-id: C8.S2.E4.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

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

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1aadec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

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

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11324 / rank
 
Normal rank
Property / Symbols used: Q11324 / qualifier
 
test:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11324 / qualifier
 
xml-id: C8.S1.XMD3.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11330 / rank
 
Normal rank
Property / Symbols used: Q11330 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11330 / qualifier
 
xml-id: C8.S1.XMD5.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11343 / rank
 
Normal rank
Property / Symbols used: Q11343 / qualifier
 
test:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11343 / qualifier
 
xml-id: C8.S1.XMD7.m1adec
Property / Symbols used
 
Property / Symbols used: Q11351 / rank
 
Normal rank
Property / Symbols used: Q11351 / qualifier
 
test:

c k ⁢ ( η ) subscript 𝑐 𝑘 𝜂 {\displaystyle{\displaystyle c_{k}(\eta)}}

c_{k}(\eta)
Property / Symbols used: Q11351 / qualifier
 
xml-id: C8.S12.XMD6.m1fdec

Latest revision as of 12:59, 2 January 2020

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DLMF:8.12.E16
No description defined

    Statements

    test
    e ± π ⁢ i ⁢ a 2 ⁢ i ⁢ sin ⁡ ( π ⁢ a ) ⁢ Q ⁡ ( - a , a ⁢ e ± π ⁢ i ) ∼ ± 1 2 - i 2 ⁢ π ⁢ a ⁢ ∑ k = 0 ∞ c k ⁢ ( 0 ) ⁢ ( - a ) - k , asymptotic-expansion superscript 𝑒 plus-or-minus 𝜋 𝑖 𝑎 2 𝑖 𝜋 𝑎 incomplete-gamma-Q 𝑎 𝑎 superscript 𝑒 plus-or-minus 𝜋 𝑖 plus-or-minus 1 2 𝑖 2 𝜋 𝑎 superscript subscript 𝑘 0 subscript 𝑐 𝑘 0 superscript 𝑎 𝑘 {\displaystyle{\displaystyle\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q% \left(-a,ae^{\pm\pi i}\right)\sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum_{k% =0}^{\infty}c_{k}(0)(-a)^{-k},}}
    0 references
    DLMF id
    DLMF:8.12.E16
    0 references
    constraint
    | ph ⁡ a | ≤ π - δ phase 𝑎 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\pi-\delta}}
    0 references
    Symbols used
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11326
    test
    Q ⁡ ( a , z ) incomplete-gamma-Q 𝑎 𝑧 {\displaystyle{\displaystyle Q\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E4.m2acdec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11324
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1hdec
    0 references
    Q11330
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1fdec
    0 references
    Q11343
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C8.S1.XMD7.m1adec
    0 references
    Q11351
    test
    c k ⁢ ( η ) subscript 𝑐 𝑘 𝜂 {\displaystyle{\displaystyle c_{k}(\eta)}}
    xml-id
    C8.S12.XMD6.m1fdec
    0 references
     
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