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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
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 . Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i 2 = − 1 {\displaystyle i^{2}=-1} along with the associative , commutative , and ...
The unit complex numbers can be realized as 2×2 real orthogonal matrices, i.e., = + [ ] = (), associating the squared modulus and complex conjugate with the determinant and transpose, respectively, of the corresponding matrix.
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.
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]
The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus | z |, which is defined as the square root of the product of z with its complex conjugate: | | = ¯. If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: | z | = x 2 + y 2 ...
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