Entire function

An entire function is complex differentiable at every point of the complex plane. Its global analyticity imposes strong restrictions on its growth and zeros.

Holomorphic function: A complex-valued function that is complex differentiable throughout an open subset of the complex plane. Entire functions are holomorphic on the whole complex plane.
Liouville's theorem: A theorem stating that every bounded entire function is constant. It shows that even a simple global growth bound can determine an entire function completely.
Complex exponential: The function defined by the complex power series sum of z to the n divided by n factorial. It is a fundamental non-polynomial entire function with no zeros.
Meromorphic function: A function holomorphic except at isolated poles in its domain. Unlike an entire function, a meromorphic function may have poles at finite complex points.
Entire function of finite order: An entire function whose maximum modulus grows no faster than an exponential of a finite power of the radius. Finite order organizes entire functions by the rate of their growth.
Complex differentiability: Differentiability of a complex-valued function with respect to a complex variable. Its definition supplies the local condition an entire function satisfies everywhere.
Cauchy's estimates: Bounds on derivatives of holomorphic functions obtained from the Cauchy integral formula. Applied on arbitrarily large circles, these estimates yield Liouville's theorem.
Sine and cosine: The trigonometric functions extended to complex arguments by their power series or exponential formulas. Both are entire and illustrate how real periodic functions extend across the complex plane.
Rational function: A quotient of two polynomials, defined wherever its denominator is nonzero. A rational function is entire only when its denominator's apparent poles cancel, leaving a polynomial.
Maximum modulus: The maximum value of a function's modulus on a specified circle or other set. Its growth as the radius increases underlies classification and factorization questions.

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