As in §24.2, let $B_n$ and $B_n\left(x\right)$ denote the $n$th Bernoulli number and polynomial, respectively, and ${\tilde{B}}_n\left(x\right)$ the $n$th Bernoulli periodic function $B_n\left(x-\lfloor x\rfloor \right)$.

Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and $f^{(2m)}(x)$ is absolutely integrable over $[a,n]$. Then

This is the Euler–Maclaurin formula. Another version is the Abel–Plana formula:

${\vartheta }_n$ being some number in the interval $(0,1)$. Sufficient conditions for the validity of this second result are:

1. (a)

   On the strip $a\le \mathrm{\Re }z\le n$, $f(z)$ is analytic in its interior, $f^{(2m)}(z)$ is continuous on its closure, and $f(z)=o\left({\mathrm{e}}^{2\pi |\mathrm{\Im }z|}\right)$ as $\mathrm{\Im }z\to \pm \mathrm{\infty }$, uniformly with respect to $\mathrm{\Re }z\in [a,n]$.

2. (b)

   $f(z)$ is real when $a\le z\le n$.

3. (c)

   The first infinite integral in (2.10.2) converges.

The formula for summation by parts is

where

This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i).

The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5.

Let $f(z)$ be analytic on the annulus , with Laurent expansion

What is the asymptotic behavior of $f_n$ as $n\to \mathrm{\infty }$ or $n\to -\mathrm{\infty }$? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion?

These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula

where $𝒞$ is a simple closed contour in the annulus that encloses $z=0$. For examples see Olver (1997b, Chapters 8, 9).

However, if $r$ is finite and $f(z)$ has algebraic or logarithmic singularities on $|z|=r$, then Darboux’s method is usually easier to apply. We need a “comparison function” $g(z)$ with the properties:

1. (a)

   $g(z)$ is analytic on .

2. (b)

   $f(z)-g(z)$ is continuous on .

3. (c)

   The coefficients in the Laurent expansion

   2.10.27		$$g(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{g}_n{z}^n,$$
   ,
   ⓘ Symbols: $r$: radious of annulus, $g(z)$: comparison function and $g_n$: coefficients Permalink: http://dlmf.nist.gov/2.10.E27 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

   have known asymptotic behavior as $n\to \pm \mathrm{\infty }$.

By allowing the contour in Cauchy’s formula to expand, we find that

Hence by the Riemann–Lebesgue lemma (§1.8(i))

This result is refinable in two important ways. First, the conditions can be weakened. It is unnecessary for $f(z)-g(z)$ to be continuous on $|z|=r$: it suffices that the integrals in (2.10.28) converge uniformly. For example, Condition (b) can be replaced by:

1. (b´)

   On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_j$, say,

   2.10.30		$$f(z)-g(z)=O\left({(z-z_j)}^{{\sigma }_j-1}\right),$$
   $z\to z_j$,
   ⓘ Symbols: $O\left(x\right)$: order not exceeding, $f(x)$: analytic function, $g(z)$: comparison function and ${\sigma }_j$: positive constant Permalink: http://dlmf.nist.gov/2.10.E30 Encodings: TeX, pMML, png See also: Annotations for §2.10(iv), §2.10 and Ch.2

   where ${\sigma }_j$ is a positive constant.

Secondly, when $f(z)-g(z)$ is $m$ times continuously differentiable on $|z|=r$ the result (2.10.29) can be strengthened. In these circumstances the integrals in (2.10.28) are integrable by parts $m$ times, yielding

Furthermore, (2.10.31) remains valid with the weaker condition

in the neighborhood of each singularity $z_j$, again with ${\sigma }_j>0$.