Editing Stirling's approximation (section)

==Versions suitable for calculators==
The approximation
<math display=block>\Gamma(z) \approx \sqrt{\frac{2 \pi}{z}} \left(\frac{z}{e} \sqrt{z \sinh\frac{1}{z} + \frac{1}{810z^6} } \right)^z</math>
and its equivalent form
<math display=block>2\ln\Gamma(z) \approx \ln(2\pi) - \ln z + z \left(2\ln z + \ln\left(z\sinh\frac{1}{z} + \frac{1}{810z^6}\right) - 2\right)</math>
can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant [[power series]] and the [[Taylor series]] expansion of the [[hyperbolic sine]] function. This approximation is good to more than 8 decimal digits for {{mvar|z}} with a real part greater than 8. Robert H. Windschitl suggested it in 2002 for computing the gamma function with fair accuracy on calculators with limited program or register memory.{{r|toth}}

Gergő Nemes proposed in 2007 an approximation which gives the same number of exact digits as the Windschitl approximation but is much simpler:{{r|Nemes2010}}
<math display=block>\Gamma(z) \approx \sqrt{\frac{2\pi}{z} } \left(\frac{1}{e} \left(z + \frac{1}{12z - \frac{1}{10z}}\right)\right)^z,</math>
or equivalently,
<math display=block> \ln\Gamma(z) \approx \tfrac{1}{2} \left(\ln(2\pi) - \ln z\right) + z\left(\ln\left(z + \frac{1}{12z - \frac{1}{10z}}\right) - 1\right). </math>

An alternative approximation for the gamma function stated by [[Srinivasa Ramanujan]] in [[Ramanujan's lost notebook]]<ref>{{citation|url=https://archive.org/details/lost-notebook/page/n337/|title=Lost Notebook and Other Unpublished Papers|first=Srinivasa|last=Ramanujan|date=14 August 1920 |author-link=Srinivasa Ramanujan|via=Internet Archive|page=339}}</ref> is
<math display=block>\Gamma(1+x) \approx \sqrt{\pi} \left(\frac{x}{e}\right)^x  \left( 8x^3 + 4x^2 + x + \frac{1}{30} \right)^{\frac{1}{6}}</math>
for {{math|''x'' ≥ 0}}. The equivalent approximation for {{math|ln ''n''!}} has an asymptotic error of {{math|{{sfrac|1|1400''n''<sup>3</sup>}}}} and is given by
<math display=block>\ln n! \approx n\ln n - n + \tfrac{1}{6}\ln(8n^3 + 4n^2 + n + \tfrac{1}{30}) + \tfrac{1}{2}\ln\pi .</math>

The approximation may be made precise by giving paired upper and lower bounds; one such inequality is{{r|E.A.Karatsuba|Mortici2011-1|Mortici2011-2|Mortici2011-3}}
<math display=block> \sqrt{\pi} \left(\frac{x}{e}\right)^x \left( 8x^3 + 4x^2 + x + \frac{1}{100} \right)^{1/6} < \Gamma(1+x) <  \sqrt{\pi} \left(\frac{x}{e}\right)^x \left( 8x^3 + 4x^2 + x + \frac{1}{30} \right)^{1/6}.</math>