In complex analysis, a partial fraction expansion is a way of writing a meromorphic function f ( z ) {\displaystyle f(z)} as an infinite sum of rational...

contour integration Morera's theorem Partial fractions in complex analysis Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. Marsden, Jerrold E.; Hoffman...

continued fractions that are rapidly convergent almost everywhere in the complex plane. The long continued fraction expression displayed in the introduction...

That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes...

rational fractions with integer coefficients. The term partial fraction is used when decomposing rational fractions into sums of simpler fractions. For example...

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour...

unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of complex analysis, and especially contour integrals, can...

Differentiation under the integral sign Trigonometric substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Trapezium...

over complex numbers. Partial fraction decomposition Partial fractions in integration Function field of an algebraic variety Algebraic fractions – a generalization...

glossary of concepts and results in real analysis and complex analysis in mathematics. In particular, it includes those in measure theory (as there is no...

to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available. The (truncated) series can be used...

In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first...

fraction [ 1 ; 2 , 2 , 2 , . . . ] {\displaystyle [1;2,2,2,...]} . This article considers only the case of periodic regular continued fractions. In other...

each ⁠ x {\displaystyle x} ⁠ in ⁠ E {\displaystyle E} ⁠ as a series of real or complex numbers. Equivalently, the partial sums s N ( x ) = ∑ n = 0 N f...

is the area of the four yellow triangles, and so on. Simplifying the fractions gives 1 + 1 4 + 1 16 + 1 64 + ⋯ , {\displaystyle 1+{\frac {1}{4}}+{\frac...

Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass factorization theorem, which asserts...

Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated...

list of articles that are considered real analysis topics. See also: glossary of real and complex analysis. Limit of a sequence Subsequential limit –...

rational approximation through continued fractions CONTINUED FRACTIONS by C. D. Olds Look up simple continued fraction in Wiktionary, the free dictionary....

In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ n = 1 ∞ 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯...

{\displaystyle \mathbb {R} ^{2}} (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists...

directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions. Here is a simple example...

{\text{for }}0<|x|<\pi .\end{aligned}}} The following continued fractions are valid in the whole complex plane: sin ⁡ x = x 1 + x 2 2 ⋅ 3 − x 2 + 2 ⋅ 3 x 2 4 ⋅...

decompression sickness. Methods used include batch mixing by partial pressure or by mass fraction, and continuous blending processes. Completed blends are...

In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function...

function around a singularity Some coefficient involved in partial fraction decomposition A remainder in modular arithmetic Residue (TV series), an English...

over the ring of the rational fractions C ( z ) . {\displaystyle \mathbb {C} (z).} If a1, ..., an are distinct complex numbers, then ea1z, ..., eanz are...

_{\mathbb {R} ^{d}}\left({\frac {\partial F_{i}}{\partial x_{j}}}(\mathbf {r} ')-{\frac {\partial F_{j}}{\partial x_{i}}}(\mathbf {r} ')\right)K(\mathbf...

condition on the partial derivatives (with respect to each yi ) at a point, the m variables yi are differentiable functions of the xj in some neighborhood...

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a...