§2.10 Sums and Sequences

Keywords:: asymptotic approximations of sums and sequences
Permalink:: http://dlmf.nist.gov/2.10

§2.10(i) Euler–Maclaurin Formula
§2.10(ii) Summation by Parts
§2.10(iii) Asymptotic Expansions of Entire Functions
§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

§2.10(i) Euler–Maclaurin Formula

Notes:: See Olver (1997b, pp. 279–292).
Keywords:: Abel–Plana formula, Euler–Maclaurin formula, asymptotic approximations of sums and sequences
Referenced by:: ¶ ‣ §24.17(i), §3.5(i), §5.17
Permalink:: http://dlmf.nist.gov/2.10.i

As in §24.2, let $\mathop{B_{{n}}\/}\nolimits$ and $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ denote the $n$ th Bernoulli number and polynomial, respectively, and $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ the $n$ th Bernoulli periodic function $\mathop{B_{{n}}\/}\nolimits\!\left(x-\left\lfloor x\right\rfloor\right)$ .

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

2.10.1 $\sum _{{j=a}}^{{n}}f(j)=\int _{{a}}^{{n}}f(x)dx+\tfrac{1}{2}f(a)+\tfrac{1}{2}f(n)+\sum _{{s=1}}^{{m-1}}\frac{\mathop{B_{{2s}}\/}\nolimits}{(2s)!}\left(f^{{(2s-1)}}(n)-f^{{(2s-1)}}(a)\right)+\int _{{a}}^{{n}}\frac{\mathop{B_{{2m}}\/}\nolimits-\mathop{\widetilde{B}_{{2m}}\/}\nolimits\!\left(x\right)}{(2m)!}f^{{(2m)}}(x)dx.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $a$ : integer, $m$ : integer, $n$ : integer and $f(x)$ : integrable function
Referenced by:: ¶ ‣ §2.10(i), ¶ ‣ §2.10(i)
Permalink:: http://dlmf.nist.gov/2.10.E1
Encodings:: TeX, pMML, png

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

2.10.2 $\sum _{{j=a}}^{{n}}f(j)=\int _{{a}}^{{n}}f(x)dx+\tfrac{1}{2}f(a)+\tfrac{1}{2}f(n)-2\int _{{0}}^{{\infty}}\frac{\imagpart{(f(a+iy))}}{e^{{2\pi y}}-1}dy+\sum _{{s=1}}^{{m}}\frac{\mathop{B_{{2s}}\/}\nolimits}{(2s)!}f^{{(2s-1)}}(n)+2\frac{(-1)^{m}}{(2m)!}\int _{{0}}^{{\infty}}\imagpart{(f^{{(2m)}}(n+i\vartheta _{n}y))}\frac{y^{{2m}}dy}{e^{{2\pi y}}-1},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\imagpart{}$ : imaginary part, $\int$ : integral, $a$ : integer, $m$ : integer, $n$ : integer and $f(x)$ : integrable function
Referenced by:: §2.10(i), ¶ ‣ §2.10(i)
Permalink:: http://dlmf.nist.gov/2.10.E2
Encodings:: TeX, pMML, png

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

(a)
On the strip $a\leq\realpart{z}\leq n$ , $f(z)$ is analytic in its interior, $f^{{(2m)}}(z)$ is continuous on its closure, and $f(z)=\mathop{o\/}\nolimits\!\left(e^{{2\pi|\imagpart{z}|}}\right)$ as $\imagpart{z}\to\pm\infty$ , uniformly with respect to $\realpart{z}\in[a,n]$ .
(b)
$f(z)$ is real when $a\leq z\leq n$ .
(c)
The first infinite integral in (2.10.2) converges.

¶ Example

Keywords:: Euler–Maclaurin formula, Glaisher’s constant, asymptotic approximations of sums and sequences

2.10.3 $S(n)=\sum _{{j=1}}^{{n}}j\mathop{\ln\/}\nolimits j$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : integer and $S(n)$ : sum
Permalink:: http://dlmf.nist.gov/2.10.E3
Encodings:: TeX, pMML, png

for large $n$ . From (2.10.1)

2.10.4 $S(n)=\tfrac{1}{2}n^{2}\mathop{\ln\/}\nolimits n-\tfrac{1}{4}n^{2}+\tfrac{1}{2}n\mathop{\ln\/}\nolimits n+\tfrac{1}{12}\mathop{\ln\/}\nolimits n+C+\sum _{{s=2}}^{{m-1}}\frac{(-\mathop{B_{{2s}}\/}\nolimits)}{2s(2s-1)(2s-2)}\frac{1}{n^{{2s-2}}}+R_{m}(n),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $m$ : integer, $n$ : integer, $S(n)$ : sum, $C$ : constant and $R_{m}(n)$ : remainder
Referenced by:: ¶ ‣ §2.10(i)
Permalink:: http://dlmf.nist.gov/2.10.E4
Encodings:: TeX, pMML, png

where $m$ ( $\geq 2$ ) is arbitrary, $C$ is a constant, and

2.10.5 $R_{m}(n)=\int _{{n}}^{{\infty}}\frac{\mathop{\widetilde{B}_{{2m}}\/}\nolimits\!\left(x\right)-\mathop{B_{{2m}}\/}\nolimits}{2m(2m-1)x^{{2m-1}}}dx.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $m$ : integer, $n$ : integer and $R_{m}(n)$ : remainder
Permalink:: http://dlmf.nist.gov/2.10.E5
Encodings:: TeX, pMML, png

From §24.12(i), (24.2.2), and (24.4.27), $\mathop{\widetilde{B}_{{2m}}\/}\nolimits\!\left(x\right)-\mathop{B_{{2m}}\/}\nolimits$ is of constant sign $(-1)^{m}$ . Thus $R_{m}(n)$ and $R_{{m+1}}(n)$ are of opposite signs, and since their difference is the term corresponding to $s=m$ in (2.10.4), $R_{m}(n)$ is bounded in absolute value by this term and has the same sign.

Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). In the present example it leads to

2.10.6 $C=\frac{\EulerConstant+\mathop{\ln\/}\nolimits\!\left(2\pi\right)}{12}-\frac{{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(2\right)}{2\pi^{2}}=\frac{1}{12}-{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-1\right),$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $C$ : constant
Permalink:: http://dlmf.nist.gov/2.10.E6
Encodings:: TeX, pMML, png

where $\EulerConstant$ is Euler’s constant (§5.2(ii)) and ${\mathop{\zeta\/}\nolimits^{{\prime}}}$ is the derivative of the Riemann zeta function (§25.2(i)). $e^{C}$ is sometimes called Glaisher’s constant. For further information on $C$ see §5.17.

Other examples that can be verified in a similar way are:

2.10.7 $\sum _{{j=1}}^{{n-1}}j^{{\alpha}}\sim\mathop{\zeta\/}\nolimits\!\left(-\alpha\right)+\frac{n^{{\alpha+1}}}{\alpha+1}\sum _{{s=0}}^{{\infty}}\binom{\alpha+1}{s}\frac{\mathop{B_{{s}}\/}\nolimits}{n^{s}},$ $n\to\infty$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $\sim$ : asymptotic equality and $n$ : integer
Referenced by:: ¶ ‣ §2.10(i), ¶ ‣ §2.10(ii)
Permalink:: http://dlmf.nist.gov/2.10.E7
Encodings:: TeX, pMML, png

where $\alpha$ ( $\neq-1$ ) is a real constant, and

2.10.8 $\sum _{{j=1}}^{{n-1}}\frac{1}{j}\sim\mathop{\ln\/}\nolimits n+\EulerConstant-\frac{1}{2n}-\sum _{{s=1}}^{{\infty}}\frac{\mathop{B_{{2s}}\/}\nolimits}{2s}\frac{1}{n^{{2s}}},$ $n\to\infty$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\EulerConstant$ : Euler’s constant, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $n$ : integer
Referenced by:: ¶ ‣ §2.10(ii)
Permalink:: http://dlmf.nist.gov/2.10.E8
Encodings:: TeX, pMML, png

In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$ , where $m$ is any positive integer satisfying $m\geq\frac{1}{2}(\alpha+1)$ .

For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004). See also Weniger (2007).

For an extension to integrals with Cauchy principal values see Elliott (1998).

§2.10(ii) Summation by Parts

Notes:: See Olver (1997b, pp. 295–299).
Keywords:: asymptotic approximations of sums and sequences, summation by parts
Permalink:: http://dlmf.nist.gov/2.10.ii

The formula for summation by parts is

2.10.9 $\sum _{{j=1}}^{{n-1}}u_{j}v_{j}=U_{{n-1}}v_{n}+\sum _{{j=1}}^{{n-1}}U_{j}(v_{j}-v_{{j+1}}),$

Symbols:: $U_{j}$ : coefficients, $u_{j}$ : terms and $v_{j}$ : terms
Permalink:: http://dlmf.nist.gov/2.10.E9
Encodings:: TeX, pMML, png

where

2.10.10 $U_{j}=u_{1}+u_{2}+\dots+u_{j}.$

Symbols:: $U_{j}$ : coefficients and $u_{j}$ : terms
Permalink:: http://dlmf.nist.gov/2.10.E10
Encodings:: TeX, pMML, png

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).

¶ Example

2.10.11 $S(\alpha,\beta,n)=\sum _{{j=1}}^{{n-1}}e^{{ij\beta}}j^{{\alpha}},$

Symbols:: $e$ : base of exponential function and $S(\alpha,\beta,n)$ : sum
Permalink:: http://dlmf.nist.gov/2.10.E11
Encodings:: TeX, pMML, png

where $\alpha$ and $\beta$ are real constants with $e^{{i\beta}}\neq 1$ .

As a first estimate for large $n$

2.10.12 $|S(\alpha,\beta,n)|\leq\sum _{{j=1}}^{{n-1}}j^{{\alpha}}=\mathop{O\/}\nolimits\!\left(1\right),\;\mathop{O\/}\nolimits\!\left(\mathop{\ln\/}\nolimits n\right),\text{ or }\mathop{O\/}\nolimits\!\left(n^{{\alpha+1}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $S(\alpha,\beta,n)$ : sum
Referenced by:: ¶ ‣ §2.10(ii)
Permalink:: http://dlmf.nist.gov/2.10.E12
Encodings:: TeX, pMML, png

according as $\alpha<-1$ , $\alpha=-1$ , or $\alpha>-1;$ see (2.10.7), (2.10.8). With $u_{j}=e^{{ij\beta}}$ , $v_{j}=j^{{\alpha}}$ ,

2.10.13 $U_{j}=e^{{i\beta}}(e^{{ij\beta}}-1)/(e^{{i\beta}}-1),$

Symbols:: $e$ : base of exponential function and $U_{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.10.E13
Encodings:: TeX, pMML, png

and

2.10.14 $S(\alpha,\beta,n)=\frac{e^{{i\beta}}}{e^{{i\beta}}-1}\left(e^{{i(n-1)\beta}}n^{{\alpha}}-1+\sum _{{j=1}}^{{n-1}}e^{{ij\beta}}\left(j^{{\alpha}}-(j+1)^{{\alpha}}\right)\right).$

Symbols:: $e$ : base of exponential function and $S(\alpha,\beta,n)$ : sum
Permalink:: http://dlmf.nist.gov/2.10.E14
Encodings:: TeX, pMML, png

Since

2.10.15 $j^{{\alpha}}-(j+1)^{{\alpha}}=-\alpha j^{{\alpha-1}}+\alpha(\alpha-1)\mathop{O\/}\nolimits\!\left(j^{{\alpha-2}}\right)$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding
Permalink:: http://dlmf.nist.gov/2.10.E15
Encodings:: TeX, pMML, png

for any real constant $\alpha$ and the set of all positive integers $j$ , we derive

2.10.16 $S(\alpha,\beta,n)=\frac{e^{{i\beta}}}{e^{{i\beta}}-1}\left(e^{{i(n-1)\beta}}n^{{\alpha}}-\alpha S(\alpha-1,\beta,n)+\mathop{O\/}\nolimits\!\left(n^{{\alpha-1}}\right)+\mathop{O\/}\nolimits\!\left(1\right)\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $e$ : base of exponential function and $S(\alpha,\beta,n)$ : sum
Referenced by:: ¶ ‣ §2.10(ii)
Permalink:: http://dlmf.nist.gov/2.10.E16
Encodings:: TeX, pMML, png

From this result and (2.10.12)

2.10.17 $S(\alpha,\beta,n)=\mathop{O\/}\nolimits\!\left(n^{{\alpha}}\right)+\mathop{O\/}\nolimits\!\left(1\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding and $S(\alpha,\beta,n)$ : sum
Permalink:: http://dlmf.nist.gov/2.10.E17
Encodings:: TeX, pMML, png

Then replacing $\alpha$ by $\alpha-1$ and resubstituting in (2.10.16), we have

2.10.18 $S(\alpha,\beta,n)=\frac{e^{{in\beta}}}{e^{{i\beta}}-1}n^{{\alpha}}+\mathop{O\/}\nolimits\!\left(n^{{\alpha-1}}\right)+\mathop{O\/}\nolimits\!\left(1\right),$ $n\to\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $e$ : base of exponential function and $S(\alpha,\beta,n)$ : sum
Permalink:: http://dlmf.nist.gov/2.10.E18
Encodings:: TeX, pMML, png

which is a useful approximation when $\alpha>0$ .

For extensions to $\alpha\leq 0$ , higher terms, and other examples, see Olver (1997b, Chapter 8).

§2.10(iii) Asymptotic Expansions of Entire Functions

Notes:: See Olver (1997b, pp. 307–309).
Keywords:: asymptotic approximations of sums and sequences, entire functions
Permalink:: http://dlmf.nist.gov/2.10.iii

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.

¶ Example

Keywords:: generalized hypergeometric function $\mathop{{{}_{{0}}F_{{2}}}\/}\nolimits$

From §§16.2(i)–16.2(ii)

2.10.19 $\mathop{{{}_{{0}}F_{{2}}}\/}\nolimits\!\left(-;1,1;x\right)=\sum _{{j=0}}^{{\infty}}\frac{x^{j}}{(j!)^{3}}.$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function and $!$ : $n!$ : factorial
Permalink:: http://dlmf.nist.gov/2.10.E19
Encodings:: TeX, pMML, png

We seek the behavior as $x\to+\infty$ . From (1.10.8)

2.10.20 $\sum _{{j=0}}^{{n-1}}\frac{x^{j}}{(j!)^{3}}=\frac{1}{2i}\int _{{\mathscr{C}}}\frac{x^{t}}{(\mathop{\Gamma\/}\nolimits\!\left(t+1\right))^{3}}\mathop{\cot\/}\nolimits\!\left(\pi t\right)dt,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral and $\mathscr{C}$ : contour
Permalink:: http://dlmf.nist.gov/2.10.E20
Encodings:: TeX, pMML, png

where $\mathscr{C}$ comprises the two semicircles and two parts of the imaginary axis depicted in Figure 2.10.1.

Figure 2.10.1: $t$ -plane. Contour $\mathscr{C}$ .

Symbols:: $\mathscr{C}$ : contour
Referenced by:: ¶ ‣ §2.10(iii)
Permalink:: http://dlmf.nist.gov/2.10.F1
Encodings:: pdf, png

From the identities

2.10.21 $\frac{\mathop{\cot\/}\nolimits\!\left(\pi t\right)}{2i}=-\frac{1}{2}-\frac{1}{e^{{-2\pi it}}-1}=\frac{1}{2}+\frac{1}{e^{{2\pi it}}-1},$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function and $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/2.10.E21
Encodings:: TeX, pMML, png

and Cauchy’s theorem, we have

2.10.22 $\sum _{{j=0}}^{{n-1}}\frac{x^{j}}{(j!)^{3}}=\int _{{-1/2}}^{{n-(1/2)}}\frac{x^{t}}{(\mathop{\Gamma\/}\nolimits\!\left(t+1\right))^{3}}dt-\int _{{\mathscr{C}_{{1}}}}\frac{x^{t}}{(\mathop{\Gamma\/}\nolimits\!\left(t+1\right))^{3}}\frac{dt}{e^{{-2\pi it}}-1}+\int _{{\mathscr{C}_{{2}}}}\frac{x^{t}}{(\mathop{\Gamma\/}\nolimits\!\left(t+1\right))^{3}}\frac{dt}{e^{{2\pi it}}-1},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral and $\mathscr{C}$ : contour
Permalink:: http://dlmf.nist.gov/2.10.E22
Encodings:: TeX, pMML, png

where $\mathscr{C}_{1},\mathscr{C}_{2}$ denote respectively the upper and lower halves of $\mathscr{C}$ . (5.11.7) shows that the integrals around the large quarter circles vanish as $n\to\infty$ . Hence

2.10.23 $\mathop{{{}_{{0}}F_{{2}}}\/}\nolimits\!\left(-;1,1;x\right)=\int _{{-1/2}}^{{\infty}}\frac{x^{t}}{(\mathop{\Gamma\/}\nolimits\!\left(t+1\right))^{3}}dt+2\realpart{\int _{{-1/2}}^{{i\infty}}\frac{x^{t}}{(\mathop{\Gamma\/}\nolimits\!\left(t+1\right))^{3}}\frac{dt}{e^{{-2\pi it}}-1}}=\int _{{0}}^{{\infty}}\frac{x^{t}}{(\mathop{\Gamma\/}\nolimits\!\left(t+1\right))^{3}}dt+\mathop{O\/}\nolimits\!\left(1\right),$ $x\to+\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
Permalink:: http://dlmf.nist.gov/2.10.E23
Encodings:: TeX, pMML, png

the last step following from $|x^{t}|\leq 1$ when $t$ is on the interval $[-\frac{1}{2},0]$ , the imaginary axis, or the small semicircle. By application of Laplace’s method (§2.3(iii)) and use again of (5.11.7), we obtain

2.10.24 $\mathop{{{}_{{0}}F_{{2}}}\/}\nolimits\!\left(-;1,1;x\right)\sim\frac{\mathop{\exp\/}\nolimits\!\left(3x^{{1/3}}\right)}{2\pi 3^{{1/2}}x^{{1/3}}},$ $x\to+\infty$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\exp\/}\nolimits z$ : exponential function and $\sim$ : asymptotic equality
Permalink:: http://dlmf.nist.gov/2.10.E24
Encodings:: TeX, pMML, png

For generalizations and other examples see Olver (1997b, Chapter 8), Ford (1960), and Berndt and Evans (1984). See also Paris and Kaminski (2001, Chapter 5) and §§16.11(i)–16.11(ii).

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

Notes:: See Olver (1997b, pp. 309–315).
Keywords:: Darboux’s method, Laurent series, Taylor series, asymptotic approximations of sums and sequences
Permalink:: http://dlmf.nist.gov/2.10.iv

Let $f(z)$ be analytic on the annulus $0<|z|<r$ , with Laurent expansion

2.10.25 $f(z)=\sum _{{n=-\infty}}^{{\infty}}f_{n}z^{n},$ $0<|z|<r$ .

Symbols:: $f(x)$ : analytic function, $f_{n}$ : coefficients and $r$ : radious of annulus
Permalink:: http://dlmf.nist.gov/2.10.E25
Encodings:: TeX, pMML, png

What is the asymptotic behavior of $f_{n}$ as $n\to\infty$ or $n\to-\infty$ ? 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

2.10.26 $f_{n}=\frac{1}{2\pi i}\int _{{\mathscr{C}}}\frac{f(z)}{z^{{n+1}}}dz,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $f(x)$ : analytic function, $f_{n}$ : coefficients and $\mathscr{C}$ : simple closed contour
Permalink:: http://dlmf.nist.gov/2.10.E26
Encodings:: TeX, pMML, png

where $\mathscr{C}$ 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:

(a)
$g(z)$ is analytic on $0<|z|<r$ .
(b)
$f(z)-g(z)$ is continuous on $0<|z|\leq r$ .
(c)

The coefficients in the Laurent expansion

2.10.27 $g(z)=\sum _{{n=-\infty}}^{{\infty}}g_{n}z^{n},$ $0<|z|<r$ ,

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

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

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

2.10.28 $f_{n}-g_{n}=\frac{1}{2\pi i}\int _{{|z|=r}}\frac{f(z)-g(z)}{z^{{n+1}}}dz=\frac{1}{2\pi r^{n}}\int _{{0}}^{{2\pi}}\left(f\left(re^{{i\theta}}\right)-g\left(re^{{i\theta}}\right)\right)e^{{-ni\theta}}d\theta.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $f(x)$ : analytic function, $f_{n}$ : coefficients, $r$ : radious of annulus, $g(z)$ : comparison function and $g_{n}$ : coefficients
Referenced by:: §2.10(iv), §2.10(iv)
Permalink:: http://dlmf.nist.gov/2.10.E28
Encodings:: TeX, pMML, png

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

2.10.29 $f_{n}=g_{n}+\mathop{o\/}\nolimits\!\left(r^{{-n}}\right),$ $n\to\pm\infty$ .

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $f_{n}$ : coefficients, $r$ : radious of annulus and $g_{n}$ : coefficients
Referenced by:: §2.10(iv)
Permalink:: http://dlmf.nist.gov/2.10.E29
Encodings:: TeX, pMML, png

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:

(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 $f(z)-g(z)=\mathop{O\/}\nolimits\!\left((z-z_{j})^{{\sigma _{j}-1}}\right),$ $z\to z_{j}$ ,

Symbols:

$\mathop{O\/}\nolimits\!\left(x\right)$ : 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

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

2.10.31 $f_{n}=g_{n}+\mathop{o\/}\nolimits\!\left(r^{{-n}}|n|^{{-m}}\right),$ $n\to\pm\infty$ .

Symbols:: $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $f_{n}$ : coefficients, $r$ : radious of annulus and $g_{n}$ : coefficients
Referenced by:: ¶ ‣ §2.10(iv), §2.10(iv)
Permalink:: http://dlmf.nist.gov/2.10.E31
Encodings:: TeX, pMML, png

Furthermore, (2.10.31) remains valid with the weaker condition

2.10.32 $f^{{(m)}}(z)-g^{{(m)}}(z)=\mathop{O\/}\nolimits\!\left((z-z_{j})^{{\sigma _{j}-1}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $f(x)$ : analytic function, $g(z)$ : comparison function and $\sigma _{j}$ : positive constant
Permalink:: http://dlmf.nist.gov/2.10.E32
Encodings:: TeX, pMML, png

in the neighborhood of each singularity $z_{j}$ , again with $\sigma _{j}>0$ .

¶ Example

Keywords:: Darboux’s method, Legendre polynomials, asymptotic approximations of sums and sequences

Let $\alpha$ be a constant in $(0,2\pi)$ and $\mathop{P_{{n}}\/}\nolimits$ denote the Legendre polynomial of degree $n$ . From §14.7(iv)

2.10.33 $f(z)\equiv\frac{1}{(1-2z\mathop{\cos\/}\nolimits\alpha+z^{2})^{{1/2}}}=\sum _{{n=0}}^{{\infty}}\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right)z^{n},$ $|z|<1$ .

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function and $f(x)$ : analytic function
Permalink:: http://dlmf.nist.gov/2.10.E33
Encodings:: TeX, pMML, png

The singularities of $f(z)$ on the unit circle are branch points at $z=e^{{\pm i\alpha}}$ . To match the limiting behavior of $f(z)$ at these points we set

2.10.34 $g(z)=e^{{-\pi i/4}}(2\mathop{\sin\/}\nolimits\alpha)^{{-1/2}}\left(e^{{-i\alpha}}-z\right)^{{-1/2}}+e^{{\pi i/4}}(2\mathop{\sin\/}\nolimits\alpha)^{{-1/2}}\left(e^{{i\alpha}}-z\right)^{{-1/2}}.$

Symbols:: $e$ : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function and $g(z)$ : comparison function
Permalink:: http://dlmf.nist.gov/2.10.E34
Encodings:: TeX, pMML, png

Here the branch of $\left(e^{{-i\alpha}}-z\right)^{{-1/2}}$ is continuous in the $z$ -plane cut along the outward-drawn ray through $z=e^{{-i\alpha}}$ and equals $e^{{i\alpha/2}}$ at $z=0$ . Similarly for $\left(e^{{i\alpha}}-z\right)^{{-1/2}}$ . In Condition (c) we have

2.10.35 $g_{n}=\left(\frac{2}{\pi\mathop{\sin\/}\nolimits\alpha}\right)^{{1/2}}\frac{\mathop{\Gamma\/}\nolimits\!\left(n+\frac{1}{2}\right)}{n!}\mathop{\cos\/}\nolimits\!\left(n\alpha+\tfrac{1}{2}\alpha-\tfrac{1}{4}\pi\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $!$ : $n!$ : factorial, $\mathop{\sin\/}\nolimits z$ : sine function and $g_{n}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.10.E35
Encodings:: TeX, pMML, png

and in the supplementary conditions we may set $m=1$ . Then from (2.10.31) and (5.11.7)

2.10.36 $\mathop{P_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right)=\left(\frac{2}{\pi n\mathop{\sin\/}\nolimits\alpha}\right)^{{1/2}}\mathop{\cos\/}\nolimits\!\left(n\alpha+\tfrac{1}{2}\alpha-\tfrac{1}{4}\pi\right)+\mathop{o\/}\nolimits\!\left(n^{{-1}}\right).$

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than and $\mathop{\sin\/}\nolimits z$ : sine function
Permalink:: http://dlmf.nist.gov/2.10.E36
Encodings:: TeX, pMML, png

For higher terms see §18.15(iii).

For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005).

For other examples and extensions see Olver (1997b, Chapter 8), Olver (1970), Wong (1989, Chapter 2), and Wong and Wyman (1974). See also Flajolet and Odlyzko (1990).

§2.10 Sums and Sequences

Contents

§2.10(i) Euler–Maclaurin Formula

¶ Example

§2.10(ii) Summation by Parts

¶ Example

§2.10(iii) Asymptotic Expansions of Entire Functions

¶ Example

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

¶ Example