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A disk with radius 1. The (open) unit disk can also be considered to be the region in the complex plane defined by {z:|z|<1}, where |z| denotes the complex modulus. (The ...

Every nonconstant entire function attains every complex value with at most one exception (Henrici 1988, p. 216; Apostol 1997). Furthermore, every analytic function assumes ...

The ordered pair (s,t), where s is the number of real embeddings of the number field and t is the number of complex-conjugate pairs of embeddings. The degree of the number ...

A theorem proved by É. Cartan in 1913 which classifies the irreducible representations of complex semisimple Lie algebras.

The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The ...

An antilinear operator A^~ satisfies the following two properties: A^~[f_1(x)+f_2(x)] = A^~f_1(x)+A^~f_2(x) (1) A^~cf(x) = c^_A^~f(x), (2) where c^_ is the complex conjugate ...

Two complex numbers z=x+iy and z^'=x^'+iy^' are added together componentwise, z+z^'=(x+x^')+i(y+y^'). In component form, (x,y)+(x^',y^')=(x+x^',y+y^') (Krantz 1999, p. 1).

The convolution of two complex-valued functions on a group G is defined as (a*b)(g)=sum_(k in G)a(k)b(k^(-1)g) where the support (set which is not zero) of each function is ...

An element of an extension field of a field F which is not algebraic over F. A transcendental number is a complex number which is transcendental over the field Q of rational ...

A proof which can be accomplished using only real numbers (i.e., real analysis instead of complex analysis; Hoffman 1998, pp. 92-93).