Holomorphic-function value at origin {Cauchy formula} is (1 / (2 * pi * i)) * contour integral (f(z)/z) * dz. Holomorphic-function value at point p is (1 / (2 * pi * i)) * contour integral (f(z)/(z - p)) * dz. Holomorphic-function nth-derivative value at origin is (n! / (2 * pi * i)) * contour integral (f(z)/z^(n + 1)) * dz.
Functions {complex function} {complex-valued function} can use complex variables: F(z), where z is complex variable. Complex function has form M + i*N, where M and N are real expressions, and i is imaginary number. For complex function F(z) = M + i*N, F(z') = M - i*N, where z' is z complex conjugate {fundamental theorem of complex numbers}.
Complex-number analytic functions {elliptic function} can be doubly periodic, be single-valued, have poles, and have singularity at infinity. Elliptic functions are power-series-equation solutions. They are elliptic-integral inverses. They are complex functions, because period ratios cannot be real numbers.
Weierstrass
Differential equations can have form dx^2 / dt^2 = A + B * x + C * x^2, where x is complex number. It has integral solutions {Weierstrass elliptic function}.
Jacobi
Differential equations can have form dx^2 / dt^2 = A + B * x + C * x^2 + D * x^3, where x is complex number. It has integral solutions {Jacobi elliptic function}.
Elliptic functions {Jacobian functions} can have higher powers. Determinants give solutions.
Darboux
Integrals {elliptic integral} {Darboux integral} can be (integral of P(x)) / (R(x))^0.5, where P is rational function, R is fourth-degree polynomial, and x is complex variable.
Abelian
Elliptic functions {Abelian integral} can have integral additions. They can also have algebraic and logarithmic terms. Integral of (R(u, z)) * dz, where f(u, z) = 0, requires more than one integral to describe domain. For Abelian integrals, equation genus is number of integrals needed to express solution {Abel's theorem}.
Complex analytic functions {holomorphic function} {holomorphic map} {differential function} {complex differentiable function} {regular function} can be differentiable for all derivatives.
equations
Complex-function real-part derivative with variable real part can equal complex-function imaginary-part derivative with variable imaginary part. Complex-function real-part derivative with variable imaginary part can equal negative of complex-function imaginary-part derivative with variable real part {Cauchy-Riemann equations, holomorphic function}. If Cauchy-Riemann equations hold, integrating on any path between two real complex-plane points obtains same result.
holomorphy
Paths can be deformable into each other or not {holomorphy}. Deformability allows canceling by going over same portion in opposite directions {homology, holomorphy}. Non-deformability does not allow canceling {homotopy, holomorphy}. If path has singularity, different paths are not homologous. At infinitesimal limit, holomorphic functions can be conformal, non-reflective, and orientation-preserving, so the only transformations are additions and multiplications. Reciprocal functions are holomorphic. Transformations (a*z + b)/(c*z + d) {bilinear transformation} {Möbius transformation, holomorphy} are holomorphic. Laplace equation in two dimensions has holomorphic function solutions.
Difference between holomorphic-function positive-frequency part and negative of negative-frequency part makes a function {hyperfunction}. Hyperfunctions have sums, derivatives, and products with analytic functions. Two hyperfunctions have no product. Hyperfunctions can represent Heaviside step function, Dirac delta function, and all analytic and holomorphic functions.
Multiple-valued complex functions {Riemann surface} can be complex-plane spirals {winding space, spiral}.
point
Riemann surfaces have a central point {branch point} about which to turn.
infinity
Surfaces can rejoin after a finite number of turns {finite order, surface} or can be infinite.
logarithm function
Logarithm-function Riemann surface is not compact but can compact to Riemann sphere.
Riemann sphere
The simplest compact/closed Riemann surface {Riemann sphere} has complex plane goes through equator. Complex plane stereographically projects onto one hemisphere, and its reciprocal projects onto other hemisphere. Circles or straight lines on complex planes are circles on spheres.
genus
Sphere has genus 0, because it has no complex moduli and has three holomorphic self-transformation parameters, for bilinear transformations. Torus has genus 1, because it has one complex modulus and one holomorphic self-transformation parameter, for translation. Genus 2 has three complex moduli and no holomorphic self-transformation parameters. Genus n has 3*n - 3 complex moduli and no holomorphic self-transformation parameters.
If Riemann hypothesis is true, functions {Riemann zeta function} can find number of primes less than N. Riemann zeta function is Dirichlet series. For complex numbers, zeta(z) = 1^-z + 2^-z + 3^-z + ..., which converges if z real part > 1, zeta(z) = 0, and z = -2, -4, -6, ... If imaginary numbers are input to zeta function, output can equal 0. Riemann zeta function equals infinity if z = 0 or 1.
Riemann hypothesis
Riemann zeta function converges if z real part = -0.5 {Riemann hypothesis} {Riemann problem}. This has no proof yet.
primes
If Riemann hypothesis is true, equation-zero locations give prime-number locations.
properties
For numbers x and N, zeta(x) = 1 + 1/2^x + 1/3^x + ... + 1/N^x. If x = 1, zeta is harmonic series. For x = 2, zeta converges to (pi)^2/6 [Euler, 1748], so sum of rational fractions gives transcendental number.
Elliptic functions can have inverses {theta function, elliptic}. Sum from z = -infinity to z = +infinity of e^(-t * n^2 + 2 * n * i * z), where n is 0 or positive integer, and t is parameter. If t > 0, theta has real part.
Multiple-valued complex functions can be sheets or spirals {winding space, function} in complex-plane Riemann surfaces.
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Date Modified: 2022.0225