DLMF:10.40.E5 (Q3583): 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.m2addec

Revision as of 12:53, 2 January 2020

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DLMF:10.40.E5
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    Statements

    test
    I ν ⁡ ( z ) ∼ e z ( 2 ⁢ π ⁢ z ) 1 2 ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ a k ⁢ ( ν ) z k ± i ⁢ e ± ν ⁢ π ⁢ i ⁢ e - z ( 2 ⁢ π ⁢ z ) 1 2 ⁢ ∑ k = 0 ∞ a k ⁢ ( ν ) z k , asymptotic-expansion modified-Bessel-first-kind 𝜈 𝑧 plus-or-minus superscript 𝑒 𝑧 superscript 2 𝜋 𝑧 1 2 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 𝑖 superscript 𝑒 plus-or-minus 𝜈 𝜋 𝑖 superscript 𝑒 𝑧 superscript 2 𝜋 𝑧 1 2 superscript subscript 𝑘 0 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 {\displaystyle{\displaystyle I_{\nu}\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{% \frac{1}{2}}}\sum_{k=0}^{\infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}}\pm ie^{\pm\nu% \pi i}\frac{e^{-z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)% }{z^{k}},}}
    0 references
    DLMF id
    DLMF:10.40.E5
    0 references
    constraint
    - 1 2 ⁢ π + δ ≤ ± ph ⁡ z ≤ 3 2 ⁢ π - δ 1 2 𝜋 𝛿 plus-or-minus phase 𝑧 3 2 𝜋 𝛿 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z% \leq\tfrac{3}{2}\pi-\delta}}
    0 references
    Symbols used
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11addec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
     
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