Search results
Results from the WOW.Com Content Network
More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .
If K is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places. If K is a function field, ν(p) = 0 for all infinite places. In the function field case, a modulus is the same thing as an effective divisor, [5] and in the number field case, a modulus can be considered as special form of Arakelov divisor. [6]
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1] χ ( a b ) = χ ( a ) χ ( b ) ; {\displaystyle \chi (ab)=\chi (a)\chi (b);} that is, χ {\displaystyle \chi } is completely multiplicative .
Complex modulus may refer to: Modulus of complex number , in mathematics, the norm or absolute value, of a complex number: | x + i y | = x 2 + y 2 {\displaystyle |x+iy|={\sqrt {x^{2}+y^{2}}}} Dynamic modulus , in materials engineering, the ratio of stress to strain under vibratory conditions
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following: The fundamental theorem of algebra. Schwarz's lemma, a result which in turn has many generalisations and applications in complex analysis. The Phragmén–Lindelöf principle, an extension to unbounded domains.
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.
In mathematics, a modular form is a (complex) analytic function on the upper half-plane, , that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory.
The most direct method of calculating a modular exponent is to calculate b e directly, then to take this number modulo m. Consider trying to compute c, given b = 4, e = 13, and m = 497: c ≡ 4 13 (mod 497) One could use a calculator to compute 4 13; this comes out to 67,108,864. Taking this value modulo 497, the answer c is determined to be 445.