View source for Complex integration, method of

''method of contour integration''

One of the universal methods in the study and applications of zeta-functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241201.png" />-functions (cf. [[Zeta-function|Zeta-function]]; [[L-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241202.png" />-function]]) and, more generally, functions defined by Dirichlet series.

The method of complex integration was first introduced by B. Riemann [[#References|[1]]] in 1876 into number theory in connection with the study of the properties of the zeta-function. The well-known modern applications of the method of complex integration, which use the Cauchy theorem on residues, the [[Phragmén–Lindelöf theorem|Phragmén–Lindelöf theorem]] on Dirichlet series, the [[Saddle point method|saddle point method]], etc., are extremely varied in their form and content. The method of complex integration is used for the analytic continuation and the derivation of functional equations of Dirichlet functions; for the derivation of approximate functional equations of these functions; for estimating the number of their non-trivial zeros and the density of the distribution of these zeros in some part of the critical strip; and for obtaining asymptotic formulas and estimates of various kinds of the most important arithmetic functions. A classical example of the method of complex integration is illustrated by the proof of the analytic continuation and the derivation of the functional equation of the Riemann zeta-function (see [[#References|[2]]], [[#References|[3]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241204.png" />, one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241205.png" /></td> </tr></table>

after summation one finds that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241206.png" />, originally defined by the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241207.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241208.png" />, is also expressed by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c0241209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Consider the integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412010.png" /></td> </tr></table>

taken along the (infinite) contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412012.png" /> pass along the lower and upper edges of the cut along the negative real axis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412013.png" />-plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412014.png" /> is a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412015.png" /> passing round the origin. The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412016.png" /> converges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412017.png" />, the convergence being uniform in any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412018.png" />, since the integrand is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412022.png" />. By Cauchy's theorem it does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412023.png" />, hence is an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412024.png" />. Assuming that on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412026.png" />, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412029.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412030.png" />, it is easy to write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412031.png" /> as a sum of integrals with respect to real variables:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412032.png" /></td> </tr></table>

In the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412033.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412034.png" />. Therefore the second term on the right-hand side of the equation is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412035.png" />, which for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412036.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412037.png" /> converges to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412038.png" />. Hence, by formula (1), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412039.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

This formula, proved under the hypothesis that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412041.png" />, provides a continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412042.png" /> to the whole plane. It is clear from this that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412043.png" /> is a single-valued analytic function on the whole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412044.png" />-plane, having as its only singularity a simple pole at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412045.png" /> with residue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412046.png" />.

To deduce the functional equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412047.png" /> it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412048.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412049.png" /> is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412050.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412051.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412052.png" /> is a contour that differs from the previous contour by joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412054.png" /> by a circular arc of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412055.png" /> with centre at the origin. The integral along the outer arc of the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412056.png" /> can be estimated in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412057.png" />, which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412058.png" /> converges to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412059.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412060.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412061.png" />. On the other hand, by the residue theorem,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412062.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412063.png" /></td> </tr></table>

Therefore, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412064.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412065.png" /></td> </tr></table>

This equation, in combination with formula (2), gives the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412066.png" /></td> </tr></table>

which, according to the theory of analytic continuation, holds throughout the whole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412067.png" />-plane. It is called the functional equation of the Riemann zeta-function.

The method of complex integration plays an important role in obtaining the approximate functional equations which lie at the basis of modern estimates of Dirichlet functions (see [[#References|[4]]], [[#References|[5]]]).

The method of complex integration is fundamental in the study of the distribution of zeros of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412069.png" />, etc. Until recently it has been applied in the form of the well-known Littlewood theorem on the number of zeros in a rectangle of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412070.png" /> that is regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412071.png" />, the Bäcklund theorem about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412072.png" />, and also to theorems about convexity of mean values of analytic functions (see [[#References|[2]]]). In 1969, H. Montgomery [[#References|[6]]] found a new, direct, more powerful way of applying the method of complex integration to these ends.

The method of complex integration in its applications to number theory naturally arises in connection with summation formulas for the coefficients of Dirichlet series (see [[#References|[2]]], [[#References|[7]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weber (ed.) , ''Riemann's gesammelte math. Werke'' , Teubner  (1892)  (Dover, reprint, 1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of the Riemann zeta-function" , Clarendon Press  (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,   "Primzahlverteilung" , Springer  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.F. Lavrik,   "Functional equations of Dirichlet functions"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31''' :  2  (1967)  pp. 431–442  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.F. Lavrik,   "Approximate functional equations of Dirichlet functions"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''32''' :  1  (1968)  pp. 134–185  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Davenport,   "Multiplicative number theory" , Springer  (1980)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.A. Karatsuba,   "A uniform estimate for the remainder term in Dirichlet's divisor problem"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''36''' :  3  (1972)  pp. 475–483  (In Russian)</TD></TR></table>



====Comments====
The derivation of the functional equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412074.png" /> can be found in [[#References|[2]]] (especially Section 2.4), in which also a number of other derivations of the functional equation are given.

The article above gives one nice illustration of the  "method of contour integration" . The philosophy behind this method is as follows. Suppose one has to evaluate an integral over a (smooth or rectifiable) contour. (A contour is a [[Curve|curve]] given by a parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412076.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412078.png" /> are continuously differentiable on each interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412079.png" /> of a finite partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412080.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412081.png" />. A contour is called smooth if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412083.png" /> do not simultaneously vanish, except at a finite number of points, see also [[Rectifiable curve|Rectifiable curve]].) The method of contour integration is to shift this contour (or extend it, like in the article above) in such a way that the integral can be easily evaluated (in most cases by the residue theorem, cf. [[Cauchy integral|Cauchy integral]]), and then to estimate the difference between the integrals over the shifted contour and the original contour (if the integrand is analytic one can sometimes use the [[Cauchy integral theorem|Cauchy integral theorem]]; if the contour has been extended, subtle estimation of the integrand might give a result). Applications of this method to certain types of integrals (e.g., of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024120/c02412086.png" />, etc.) and in the proof of many important theorems in complex analysis can be found in almost any textbook on complex analysis, e.g. [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]]. Reference [[#References|[a5]]] contains a wealth of formulas derived by the method of contour integration.

A novel approach to the prime number theorem using the method of contour integration was given in [[#References|[a1]]].

The method of contour integration is also called the contour integral method.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Newman,   "Simple analytic proof of the prime number theorem"  ''Amer. Math. Monthly'' , '''11'''  (1980)  pp. 693–696</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bak,   D.J. Newman,   "Complex analysis" , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.V. Ahlfors,   "Complex analysis" , McGraw-Hill  (1979)  pp. 241</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.A. Evgrafov,   "Analytic functions" , Dover, reprint , Philadelphia  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.S. Mitrinović,   J.D. Kečkić,   "The Cauchy method of residues: theory and applications" , Reidel  (1984)</TD></TR></table>