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Encyclopedia > Entire function

In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. Typical examples of entire functions are the polynomials, the exponential function, and sums, products and compositions of these. Every entire function can be represented as a power series which converges everywhere. Neither the natural logarithm nor the square root function is entire. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Partial plot of a function f. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... The exponential function is one of the most important functions in mathematics. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... The natural logarithm is the logarithm to the base e, where e is equal to 2. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or...

Note that an entire function may have a singularity or even an essential singularity at the complex point at infinity. In the latter case, it is called a transcendental entire function. In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... In complex analysis, an essential singularity of a function is a severe singularity near which the function exhibits extreme behavior. ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...

Liouville's theorem establishes an important property of entire functions — an entire function which is bounded must be constant. This property can be used for an elegant proof of the fundamental theorem of algebra. Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value 0. Liouvilles theorem in complex analysis states that every bounded (i. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In complex analysis, mathematician Charles Ã‰mile Picards name is given to two theorems regarding the range of an analytic function. ... The exponential function is one of the most important functions in mathematics. ... Results from FactBites:

 Entire function - Wikipedia, the free encyclopedia (206 words) In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. Neither the natural logarithm nor the square root function is entire. Note that an entire function may have a singularity or even an essential singularity at the complex point at infinity.
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