+9589 I am totally confused when I am on Wikipedia...... It just gives me a lot of nouns.......
I have a few questions:
1) What is a holomorphic function?
2) What is a elliptic function?
3) What is Legendre Symbol?
4) What is a quadratic residue modulo?
Hello Max: Here are some "technical definition" of the above:
1) What is a holomorphic function?
"A synonym for analytic function, regular function, differentiable function, complex differentiable function, and holomorphic map. The word derives from the Greek ολος (holos), meaning "whole, " and μορφη (morphe), meaning "form" or "appearance."
Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic function", while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts."
2)What is a elliptic function?
"A doubly periodic function with periods 2ω_1 and 2ω_2 such that
f(z + 2ω_1) = f(z + 2ω_2) = f(z), which is analytic and has no singularities except for poles in the finite part of the complex plane. The half-period ratio τ congruent ω_2/ω_1 must not be purely real, because if it is, the function reduces to a singly periodic function if τ is rational, and a constant if τ is irrational. ω_1 and ω_2 are labeled such that ℑ[τ] congruent ℑ[ω_2/ω_1]>0, where ℑ[z] is the imaginary part."
3) What is Legendre Symbol?
"The Legendre symbol is a number theoretic function (a/p) which is defined to be equal to ± 1 depending on whether a is a quadratic residue modulo p. The definition is sometimes generalized to have value 0 if p|a, (a/p) = (a|p) congruent {0 | if p|a
1 | if a is a quadratic residue modulo p
-1 | if a is a quadratic nonresidue modulo p. auto right match
If p is an odd prime, then the Jacobi symbol reduces to the Legendre symbol. The Legendre symbol is implemented in the Wolfram Language via the Jacobi symbol, JacobiSymbol[a, p]."
4) What is a quadratic residue modulo?
"If there is an integer 0<x<p such that
x^2 congruent q (mod p), i.e., the congruence (-1) has a solution, then q is said to be a quadratic residue (mod p). Note that the trivial case q = 0 is generally excluded from lists of quadratic residues so that the number of quadratic residues (mod n) is taken to be one less than the number of squares (mod n). However, other sources include 0 as a quadratic residue."
Hi Max, put simply,a holomorphic function is one that represents a surface without any punctures or holes in it.(You will come across a few 'punctured ' surfaces if you study complex analysis) So it's differentiable at every point and it's continuous. Think about them the same way you think about lines and curves,if they have cuts or breaks in them,they are not continuous, holomorphic functions just extend that idea onto surfaces in space.
