DLMF:8.11.E6 (Q2577): Difference between revisions

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Property / constraint
 

| ph ⁡ a | ≤ π 2 - δ ph 𝑎 𝜋 2 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta}}

|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta
Property / constraint: | ph ⁡ a | ≤ π 2 - δ ph 𝑎 𝜋 2 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
test:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11adec
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.m2abdec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: incomplete gamma function / rank
 
Normal rank
Property / Symbols used: incomplete gamma function / qualifier
 
test:

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

\incgamma@{\NVar{a}}{\NVar{z}}
Property / Symbols used: incomplete gamma function / qualifier
 
xml-id: C8.S2.E1.m2aadec
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.m1abdec
Property / Symbols used
 
Property / Symbols used: Q11332 / rank
 
Normal rank
Property / Symbols used: Q11332 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11332 / qualifier
 
xml-id: C8.S1.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q11323 / rank
 
Normal rank
Property / Symbols used: Q11323 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11323 / qualifier
 
xml-id: C8.S1.XMD2.m1ddec
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.m1edec
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.m1cdec
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.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11344 / rank
 
Normal rank
Property / Symbols used: Q11344 / qualifier
 
test:

λ 𝜆 {\displaystyle{\displaystyle\lambda}}

\lambda
Property / Symbols used: Q11344 / qualifier
 
xml-id: C8.S11.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q11345 / rank
 
Normal rank
Property / Symbols used: Q11345 / qualifier
 
test:

b k ⁢ ( λ ) subscript 𝑏 𝑘 𝜆 {\displaystyle{\displaystyle b_{k}(\lambda)}}

b_{k}(\lambda)
Property / Symbols used: Q11345 / qualifier
 
xml-id: C8.S11.XMD4.m1dec

Latest revision as of 12:54, 2 January 2020

No description defined
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English
DLMF:8.11.E6
No description defined

    Statements

    test
    γ ⁡ ( a , z ) ∼ - z a ⁢ e - z ⁢ ∑ k = 0 ∞ ( - a ) k ⁢ b k ⁢ ( λ ) ( z - a ) 2 ⁢ k + 1 , asymptotic-expansion incomplete-gamma 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 superscript subscript 𝑘 0 superscript 𝑎 𝑘 subscript 𝑏 𝑘 𝜆 superscript 𝑧 𝑎 2 𝑘 1 {\displaystyle{\displaystyle\gamma\left(a,z\right)\sim-z^{a}e^{-z}\sum_{k=0}^{% \infty}\frac{(-a)^{k}b_{k}(\lambda)}{(z-a)^{2k+1}},}}
    0 references
    DLMF id
    DLMF:8.11.E6
    0 references
    constraint
    | ph ⁡ a | ≤ π 2 - δ phase 𝑎 𝜋 2 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta}}
    0 references
    0 < λ < 1 0 𝜆 1 {\displaystyle{\displaystyle 0<\lambda<1}}
    0 references
    | ph ⁡ a | ≤ π 2 - δ ph 𝑎 𝜋 2 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta}}
    0 references
    Symbols used
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    incomplete gamma function
    test
    γ ⁡ ( a , z ) incomplete-gamma 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E1.m2aadec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
    0 references
    Q11332
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S1.XMD1.m1dec
    0 references
    Q11323
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1ddec
    0 references
    Q11324
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1edec
    0 references
    Q11330
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1cdec
    0 references
    Q11343
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C8.S1.XMD7.m1bdec
    0 references
    Q11344
    test
    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
    xml-id
    C8.S11.XMD3.m1dec
    0 references
    Q11345
    test
    b k ⁢ ( λ ) subscript 𝑏 𝑘 𝜆 {\displaystyle{\displaystyle b_{k}(\lambda)}}
    xml-id
    C8.S11.XMD4.m1dec
    0 references
     
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