Search results
Results from the WOW.Com Content Network
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g.More precisely, given an open set in the complex plane and a function :, the antiderivative of is a function : that satisfies =.
Johannes Blümlein and Carsten Schneider (Eds.): Anti-Differentiation and the Calculation of Feynman Amplitudes, Springer, ISBN 978-3-030-80218-9 (2021). Stefan Weinzierl: Feynman Integrals: A Comprehensive Treatment for Students and Researchers, Springer, ISBN 978-3-030-99560-7 (Jun., 2023).
Two other well-known examples are when integration by parts is applied to a function expressed as a product of 1 and itself. This works if the derivative of the function is known, and the integral of this derivative times is also known. The first example is (). We write this as:
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.
All such algorithms proceed in two steps: The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point.
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.
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. [1] [2] [3]Contour integration is closely related to the calculus of residues, [4] a method of complex analysis.