Concept
Cauchy's integral formula
Related concepts (16)
Morera's theorem
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic. Morera's theorem states that a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies for every closed piecewise C1 curve in D must be holomorphic on D. The assumption of Morera's theorem is equivalent to f locally having an antiderivative on D. The converse of the theorem is not true in general.
Winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise.
Meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function. The term comes from the Greek meros (μέρος), meaning "part". Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: any pole must coincide with a zero of the denominator.
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.
Complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.