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A bounded entire function in the complex plane C is constant. The fundamental theorem of algebra follows as a simple corollary.

Any entire analytic function whose range omits two points must be a constant function. Of course, an entire function that omits a single point from its range need not be a ...

The duplication formula for Rogers L-function follows from Abel's functional equation and is given by 1/2L(x^2)=L(x)-L(x/(1+x)).

An anti-analytic function is a function f satisfying the condition (df)/(dz)=0. (1) Using the result ...

A Bergman kernel is a function of a complex variable with the "reproducing kernel" property defined for any domain in which there exist nonzero analytic functions of class ...

If a is a point in the open unit disk, then the Blaschke factor is defined by B_a(z)=(z-a)/(1-a^_z), where a^_ is the complex conjugate of a. Blaschke factors allow the ...

Let f be a bounded analytic function on D(0,1) vanishing to order m>=0 at 0 and let {a_j} be its other zeros, listed with multiplicities. Then ...

A Blaschke product is an expression of the form B(z)=z^mproduct_(j=1)^infty-(a^__j)/(|a_j|)B_(a_j)(z), where m is a nonnegative integer and z^_ is the complex conjugate.

A linear operator A:D(A)->H from its domain D(A) into a Hilbert space H is closed if for any sequence of vectors v_n in D(A) such that v_n->v and Av_n->x as n->infty, it ...

A function whose range is in the complex numbers is said to be a complex function, or a complex-valued function.