百科问答小站 logo
百科问答小站 font logo



什么是「斯特林公式」? 第1页

  

user avatar   miaplacidus-official 网友的相关建议: 
      

The Stirling's formula is an approximation to the factorial function and can be generalised for the gamma function; it states that which means that the two quantities are asymptotically equal when approaches infinity and can be used for numerical estimation.

Proof:

First, we rewrite the factorial function using Euler's integral of the second kind (definition for the gamma function )as and change the variable of integration to such that :

Let us define , then there is . For , there is while we may apply logaritihm on both sides to obtain . Furthermore, for , from the Mercator series we have the asymptotic approximation for the natural logarithm , which may be use to evaluate . Therefore, we have shown the approximation as .

Then, let us consider which implies there is . Since is integrable on , from the Lebesgue's dominated convergence theorem we can deduce that is also integrable on and hence can be properly defined and evaluated.

In summary, we have shown that for there is the asymptotic equality which leads to . From the Gaussian integral and the aforementioned Euler's integral representation of , we have finally proved the correctness of the Stirling's approximation as .


The Stirling's approximation can be further applied for the gamma function for where . Since the error term is large for , we may use the reflection formula to estimate .

The Stirling's approximation can also be generalised to the asymptotic expansion named Stirling series for both the factorial and the gamma function to an arbitrary-precision.

Notice that the series is not absolutely convergent; for any particular , only a finite number of terms can be used to ameliorate the accuracy of estimation.




  

相关话题

  实数在自然界有用吗? 
  有哪些无用的论文、研究? 
  太阳辐射模拟光源全球有哪些研发、制造企业? 
  《从一到正无穷》第三章的空间想象是什么意思? 
  非常硬核的数学题,大家能否解出? 
  运用复数证明平面几何的原理有哪些? 
  确定正弦函数图象需要几个点? 
  怎么样才能学德好数学? 
  算子代数是一门怎样的数学分支?学习算子代数需要怎样的基础? 
  氢原子外层电子跃迁过程中,其能量是上能级还是下能级所对应能量? 

前一个讨论
男孩比女孩更擅长数学是真的吗?
下一个讨论
为什么春晚的《时间都去哪儿了》非得用30年父女合影作为背景,而不是相对论一类的物理学内容?





© 2025-06-07 - tinynew.org. All Rights Reserved.
© 2025-06-07 - tinynew.org. 保留所有权利