DLMF:9.7.E12 (Q2819): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.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.m1agdec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: Q11385 / rank
 
Normal rank
Property / Symbols used: Q11385 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11385 / qualifier
 
xml-id: C9.S1.XMD1.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11381 / rank
 
Normal rank
Property / Symbols used: Q11381 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11381 / qualifier
 
xml-id: C9.S1.XMD3.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11386 / rank
 
Normal rank
Property / Symbols used: Q11386 / qualifier
 
test:

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

\delta
Property / Symbols used: Q11386 / qualifier
 
xml-id: C9.S1.XMD4.m1gdec
Property / Symbols used
 
Property / Symbols used: change of variable (locally) / rank
 
Normal rank
Property / Symbols used: change of variable (locally) / qualifier
 
test:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: change of variable (locally) / qualifier
 
xml-id: C9.S7.XMD1.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11388 / rank
 
Normal rank
Property / Symbols used: Q11388 / qualifier
 
test:

v s subscript 𝑣 𝑠 {\displaystyle{\displaystyle v_{s}}}

v_{s}
Property / Symbols used: Q11388 / qualifier
 
xml-id: C9.S7.XMD3.m1ddec

Latest revision as of 13:34, 2 January 2020

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English
DLMF:9.7.E12
No description defined

    Statements

    test
    Bi ′ ⁡ ( - z ) ∼ z 1 / 4 π ⁢ ( cos ⁡ ( ζ - 1 4 ⁢ π ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ v 2 ⁢ k ζ 2 ⁢ k + sin ⁡ ( ζ - 1 4 ⁢ π ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ v 2 ⁢ k + 1 ζ 2 ⁢ k + 1 ) , asymptotic-expansion diffop Airy-Bi 1 𝑧 superscript 𝑧 1 4 𝜋 𝜁 1 4 𝜋 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑣 2 𝑘 superscript 𝜁 2 𝑘 𝜁 1 4 𝜋 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑣 2 𝑘 1 superscript 𝜁 2 𝑘 1 {\displaystyle{\displaystyle\mathrm{Bi}'\left(-z\right)\sim\frac{z^{1/4}}{% \sqrt{\pi}}\left(\cos\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)% ^{k}\frac{v_{2k}}{\zeta^{2k}}+\sin\left(\zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}% ^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}}\right),}}
    0 references
    DLMF id
    DLMF:9.7.E12
    0 references
    constraint
    | ph ⁡ z | ≤ 2 3 ⁢ π - δ phase 𝑧 2 3 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta}}
    0 references
    Symbols used
    Q11081
    test
    Bi ⁡ ( z ) Airy-Bi 𝑧 {\displaystyle{\displaystyle\mathrm{Bi}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m2acdec
    0 references
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11385
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C9.S1.XMD1.m1hdec
    0 references
    Q11381
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1hdec
    0 references
    Q11386
    test
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C9.S1.XMD4.m1gdec
    0 references
    change of variable (locally)
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S7.XMD1.m1hdec
    0 references
    Q11388
    test
    v s subscript 𝑣 𝑠 {\displaystyle{\displaystyle v_{s}}}
    xml-id
    C9.S7.XMD3.m1ddec
    0 references
     
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