In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions...

) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. The residue of...

plane application of the Cauchy integral formula application of the residue theorem One method can be used, or a combination of these methods, or various...

{\textstyle {\overline {U}}} . The Cauchy integral theorem leads to Cauchy's integral formula and the residue theorem. If one assumes that the partial derivatives...

cubic residue modulo q. Let q = 3n + 2; since 0 = 03 is obviously a cubic residue, assume x is not divisible by q. Then by Fermat's little theorem, x q...

locating residues when one applies Cauchy's residue theorem. Rouché's theorem can also be used to give a short proof of the fundamental theorem of algebra...

the residue classes modulo a prime number are a field. See the article prime field for more details. Because the modulus is prime, Lagrange's theorem applies:...

D \ {0}. In the special case where the residue of g at 0 is zero the conjecture follows from the "Great Picard's Theorem". Elsner, B. (1999). "Hyperelliptic...

important role throughout complex analysis (cf. the statement of the residue theorem). In the context of complex analysis, the winding number of a closed...

and so no other residues. By the residue theorem we have that the integral about C is the product of 2πi and the sum of the residues. Together, the sum...

{\displaystyle a} is a quadratic residue, which can be checked using the law of quadratic reciprocity. The quadratic reciprocity theorem was conjectured by Leonhard...

function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable...

Cauchy–Riemann equations Methods of contour integration Residue (complex analysis) Mittag-Leffler's theorem Ahlfors, Lars (January 1, 1979), Complex Analysis...

the residue theorem, which is a result for meromorphic functions, and a related result, the argument principle. It is known from Morera's theorem that...

these residue classes, and its powers a, a2, ... , ak modulo n form a subgroup of the group of residue classes, with ak ≡ 1 (mod n). Lagrange's theorem says...

In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers...

(complex analysis) Marden's theorem Nyquist stability criterion Pole–zero plot Residue (complex analysis) Rouché's theorem Sendov's conjecture Conway,...

least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one representative of each residue class...

In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded...

important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the function be isolated...

In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively...

the same representation in the residue numeral system defined by the mis. More precisely, the Chinese remainder theorem asserts that each of the M different...

is simply zero, or in case the region includes singularities, the residue theorem computes the integral in terms of the singularities. This also implies...

In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number...

j-invariant. Least-squares spectral analysis Multidimensional transform Residue theorem integrals of f(z), singularities, poles Sine and cosine transforms...

theorists of the 17th and 18th centuries established theorems and formed conjectures about quadratic residues, but the first systematic treatment is § IV of...

is no obvious general rule at work. He goes on to say The theorems on biquadratic residues gleam with the greatest simplicity and genuine beauty only...

The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil (and any two-dimensional body including...

Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is...

{1}{z}}\right)dz} at z = 0 {\displaystyle z=0} . Riemann sphere Algebraic variety Residue theorem Michèle Audin, Analyse Complexe, lecture notes of the University of...