Stirling's Formula for Gamma Function
(Redirected from Stirling's Asymptotic Series)
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Theorem
Let $\Gamma$ denote the Gamma function.
Then:
- $\map \Gamma {x + 1} = \sqrt {2 \pi x} \, x^x e^{-x} \paren {1 + \dfrac 1 {12 x} + \dfrac 1 {288 x^2} - \dfrac {139} {51 \, 480 x^3} + \cdots}$
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Proof
This theorem requires a proof. In particular: Use Binet's Formula for Logarithm of Gamma Function/Formulation 2. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also known as
This formula is also known as Stirling's asymptotic series.
Source of Name
This entry was named for James Stirling.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $16.15$: Asymptotic Expansions for the Gamma Function