DLMF:2.6.E52 (Q878): 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.m2afdec

Revision as of 16:18, 1 January 2020

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DLMF:2.6.E52
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    Statements

    test
    I μ ⁢ f ⁢ ( x ) = ∑ s = 0 n - 1 ( - 1 ) s ⁢ Γ ⁡ ( 1 - s - α ) Γ ⁡ ( μ + 1 - s - α ) ⁢ x μ - s - α - π sin ⁡ ( π ⁢ α ) ⁢ ∑ s = 1 n 1 Γ ⁡ ( μ + 1 - s ) ⁢ x μ - s ( s - 1 ) ! + 1 x n ⁢ δ n ⁢ ( x ) . superscript 𝐼 𝜇 𝑓 𝑥 superscript subscript 𝑠 0 𝑛 1 superscript 1 𝑠 Euler-Gamma 1 𝑠 𝛼 Euler-Gamma 𝜇 1 𝑠 𝛼 superscript 𝑥 𝜇 𝑠 𝛼 𝜋 𝜋 𝛼 superscript subscript 𝑠 1 𝑛 1 Euler-Gamma 𝜇 1 𝑠 superscript 𝑥 𝜇 𝑠 𝑠 1 1 superscript 𝑥 𝑛 subscript 𝛿 𝑛 𝑥 {\displaystyle{\displaystyle I^{\mu}f(x)=\sum_{s=0}^{n-1}(-1)^{s}\frac{\Gamma% \left(1-s-\alpha\right)}{\Gamma\left(\mu+1-s-\alpha\right)}x^{\mu-s-\alpha}-% \frac{\pi}{\sin\left(\pi\alpha\right)}\sum_{s=1}^{n}\frac{1}{\Gamma\left(\mu+1% -s\right)}\frac{x^{\mu-s}}{(s-1)!}+\frac{1}{x^{n}}\delta_{n}(x).}}
    0 references
    DLMF id
    DLMF:2.6.E52
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2akdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
     
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