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First derivative test; Second derivative test; Extreme value theorem; Differential equation; Differential operator; Newton's method; Taylor's theorem; L'Hôpital's rule; General Leibniz rule; Mean value theorem; Logarithmic derivative; Differential (calculus) Related rates; Regiomontanus' angle maximization problem; Rolle's theorem
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators [1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives ...
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
Color wheel graph of the function f(x) = (x 2 − 1)(x − 2 − i) 2 / x 2 + 2 + 2i . Hue represents the argument , brightness the magnitude. One of the central tools in complex analysis is the line integral .
The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum.
Geometric calculus defines a derivative operator ∇ = ê i ∂ i under its geometric product — that is, for a k-vector field ψ(r), the derivative ∇ψ generally contains terms of grade k + 1 and k − 1. For example, a vector field (k = 1) generally has in its derivative a scalar part, the divergence (k = 0), and a bivector part, the curl ...
The Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic property are pure topological; [80] [39] [31] Serre called this the Oka principle. [84] They are now posed, and solved, for arbitrary complex manifold M, in terms of conditions on M.
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 (is analytic). Holomorphic functions are the central objects of study in complex analysis.