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Stirling's formula is usually stated in the form $\log \Gamma(s) = (s-\frac12) \log{s} - s + \log\sqrt{2\pi} + E(s)$, where $E(s) = c_1/s + c_2/s^2 + \dots + O(s^{-K})$ for certain absolute constants $c_i$. I am interested in having a uniform approximation for $E(s)$ that is valid for all $s = \sigma + it$ with $\sigma>0$ fixed and $|t| \leq X$ for $X \geq 1$. Does there exist a known "nice" approximation for $E(s)$ of the form $E(s) = F(s) + O(X^{-K})$, where $F(s)$, which depends on $K$ and $X$ of course, has an explicit shape? Bonus points for explicit bounds
on $F(s)$ and its derivatives uniformly valid in $|t| \leq X$.

EDIT ADDED July 28 2010: I am doubtful if there is a positive answer to my question. As a simple example, consider the rate of convergence of the Taylor series of the cosine function. Of course, $\cos(x) = 1- \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \pm \frac{x^{2k}}{(2k)!} + R_{2k}(x)$ where $R_{2k}(x) = \cos^{(2k+1)}(\xi) \frac{x^{2k+1}}{(2k+1)!}$ for some $|\xi| \leq |x|$. In order to get an error term that is $O(X^{-K})$ uniformly for $|x| \leq X$ we need to take $k$ roughly on the order of $X$ (since that is when the factorial in the denominator wins over the size of $X^{2k+1}$); at this point the error term gets very small very fast . This is a lot of terms!

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Stirling's formula is usually stated in the form $\log \Gamma(s) = (s-\frac12) \log{s} - s + \log\sqrt{2\pi} + E(s)$, where $E(s) = c_1/s + c_2/s^2 + \dots + O(s^{-K})$ for certain absolute constants $c_i$. I am interested in having a uniform approximation for $E(s)$ that is valid for all $s = \sigma + it$ with $\sigma>0$ fixed and $|t| \leq X$ for $X \geq 1$. Does there exist a known "nice" approximation for $E(s)$ of the form $E(s) = F(s) + O(X^{-K})$, where $F(s)$, which depends on $K$ and $X$ of course, has an explicit shape? Bonus points for explicit bounds
on $F(s)$ and its derivatives uniformly valid in $|t| \leq X$.

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Stirling's formula is usually stated in the form $\log \Gamma(s) = (s-\frac12) \log{s} - s + \log\sqrt{2\pi} + E(s)$, where $E(s) = c_1/s + c_2/s^2 + \dots + O(s^{-K})$ for certain absolute constants $c_i$. I am interested in having a uniform approximation for $E(s)$ that is valid for all $s = \sigma + it$ with $\sigma$ \sigma>0$fixed and$|t| \leq X$for$X \geq 1$. Does there exist a known "nice" approximation for$E(s)$of the form$E(s) = F(s) + O(X^{-K})$, where$F(s)$, which depends on$K$and$X$of course, has an explicit shape? Bonus points for explicit bounds on$F(s)$and its derivatives uniformly valid in$|t| \leq X\$.

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