Editing Double factorial (section)

==Additional identities==
For integer values of {{mvar|n}},
<!-- FORMER TEXT Using the extension of the double factorial to even arguments, for even values of ''n'', -->
<math display="block">\int_{0}^\frac{\pi}{2}\sin^n x\,dx=\int_{0}^\frac{\pi}{2}\cos^n x\,dx=\frac{(n-1)!!}{n!!}\times
\begin{cases}1 & \text{if } n \text{ is odd} \\ \frac{\pi}{2} & \text{if } n \text{ is even.}\end{cases}</math>
Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is
<math display="block">\int_{0}^\frac{\pi}{2}\sin^n x\,dx=\int_{0}^\frac{\pi}{2}\cos^n x\,dx=\frac{(n-1)!!}{n!!} \sqrt{\frac{\pi}{2}}\,.</math>

Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.<ref name="meserve"/><ref>{{cite journal
 | last1 = Dassios | first1 = George | author1-link = George Dassios
 | last2 = Kiriaki | first2 = Kiriakie
 | issue = A
 | journal = Bulletin de la Société Mathématique de Grèce
 | mr = 935868
 | pages = 40–43
 | title = A useful application of Gauss theorem
 | volume = 28
 | year = 1987}}</ref>

Double factorials of odd numbers are related to the [[gamma function]] by the identity:

<math display="block">(2n-1)!! = 2^n \cdot \frac{\Gamma\left(\frac{1}{2} + n\right)} {\sqrt{\pi}} = (-2)^n \cdot \frac{\sqrt{\pi}} { \Gamma\left(\frac{1}{2} - n\right)}\,.</math>

Some additional identities involving double factorials of odd numbers are:<ref name="callan"/>

<math display="block">\begin{align}
(2n-1)!! &= \sum_{k=0}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!! \\
  &= \sum_{k=1}^{n} \binom{n}{k} (2k-3)!! (2(n-k)-1)!! \\
  &= \sum_{k=0}^{n} \binom{2n-k-1}{k-1} \frac{(2k-1)(2n-k+1)}{k+1}(2n-2k-3)!! \\
  &= \sum_{k=1}^{n} \frac{(n-1)!}{(k-1)!} k(2k-3)!!\,.
\end{align}</math>

An approximation for the ratio of the double factorial of two consecutive integers is
<math display="block"> \frac{(2n)!!}{(2n-1)!!} \approx \sqrt{\pi n}. </math>
This approximation gets more accurate as {{mvar|n}} increases, which can be seen as a result of the [[Wallis%27_integrals#Deducing the Double Factorial Ratio | Wallis Integral]].