contour integral

Evaluation of the inverse z transform using complex series and contour integrals.

The Legendre functions can be written as contour integrals.

Contour integrals involving the extension of F clearly split into two, using part of the real axis.

Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

Path dependence, path independence, contour integrals.

One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.

Siegel derived it from the Riemann-Siegel integral formula, an expression for the zeta function involving contour integrals.

They satisfy certain partial differential equations, and can also be given in terms of Euler-type integrals and contour integrals.

It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals.

This type of path for contour integrals was first used by Hermann Hankel in his investigations of the Gamma function.

In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals.

Similarly by converting the integration paths to contour integrals one can obtain other formulas for the eta function, such as this generalisation (Milgram, 2013) valid for and all :

The residue theorem allows contour integrals to be used in the complex plane to find integrals of real-valued functions of a real variable (see residue theorem for an example).

More functions yet, including the hypergeometric function and special cases thereof, can be represented by means of complex contour integrals of products and quotients of the gamma function, called Mellin-Barnes integrals.

When it comes to taking contour integrals, moving basepoint from P to another choice Q makes no difference to the result, since there will be cancellation of integrals from P to Q and back.

We will evaluate it by expressing it as a limit of contour integrals along the contour 'C' that goes along the real number line from −'a' to 'a' and then counterclockwise along a semicircle centered at 0 from 'a' to −'a'.

In joint work with A-K Kassam we have found that a fourth-order Runge-Kutta scheme known as ETDRK4, developed by Cox and Matthews, performs impressively if its coefficients are stably computed by means of contour integrals in the complex plane.