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In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
Two volumes of the Bateman Manuscript are specific to integral transforms. There are several web sites which have tables of integrals and integrals on demand. Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration.
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space ), which the Mathematics Subject ...
The Laurent series for a complex function () about a point is given by = = (), where and are constants, with defined by a contour integral that generalizes Cauchy's integral formula: = () +. The path of integration γ {\displaystyle \gamma } is counterclockwise around a Jordan curve enclosing c {\displaystyle c} and lying in an annulus A ...
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse .
In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy -plane bounded by the graph of f , the x -axis, and the lines x = a and x = b , such that area above the x -axis adds to the total, and that below the x -axis subtracts from the total.
When the Abel–Plana formula is applied to the defining series of the polylogarithm, a Hermite-type integral representation results that is valid for all complex z and for all complex s: = + (, ) () + ( ) (+) / where Γ is the upper incomplete gamma-function.
Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics, the exponential integral Ei is a special function on the complex plane .
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