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In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function ''f''(''z'') has a root at ''w'', then ''f''(''z'')/(''z−w'') is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. ==Properties== Every entire function ''f''(''z'') can be represented as a power series : that converges everywhere in the complex plane, hence uniformly on compact sets. The radius of convergence is infinite, which implies that : or : Any power series satisfying this criterion will represent an entire function. If (and only if) the coefficients of the power series are all real then the function (obviously) takes real values for real arguments, and the value of the function at the complex conjugate of ''z'' will be the complex conjugate of the value at ''z''. Such functions are sometimes called self-conjugate (the conjugate function, , being given by 〔See for example 〕 If the real part of an entire function is known in a neighborhood of a point then both the real and imaginary parts are known for the whole complex plane, up to an imaginary constant. For instance, if the real part is known in a neighborhood of zero, then we can find the coefficients for ''n'' > 0 from the following derivatives with respect to a real variable ''r'': : : (Likewise, if the imaginary part is known in a neighborhood then the function is determined up to a real constant.) In fact, if the real part is known just on an arc of a circle, then the function is determined up to an imaginary constant. (For instance, if it the real part is known on part of the unit circle, then it is known on the whole unit circle by analytic extension, and then the coefficients of the infinite series are determined from the coefficients of the Fourier series for the real part on the unit circle.) Note however that an entire function is ''not'' determined by its real part on all curves. In particular, if the real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can be added to the function we are trying to determine. For example, if the curve where the real part is known is the real line, then we can add ''i'' times any self-conjugate function. If the curve forms a loop, then the only functions whose real part is zero on the curve are those that are everywhere equal to some imaginary number. The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes (or "roots"). The entire functions on the complex plane form an integral domain (in fact a Prüfer domain). They also form a commutative unital associative algebra over the complex numbers. Liouville's theorem states that any bounded entire function must be constant. Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra. As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere (complex plane ''and'' the point at infinity) is constant. Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function. Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function ''f'' and any complex ''w'' there is a sequence with and . Picard's little theorem is a much stronger result: any non-constant entire function takes on every complex number as value, possibly with a single exception. When an exception exists, it is called a lacunary value of the function. The possibility of a lacunary value is illustrated by the exponential function, which never takes on the value 0. One can take a logarithm of an entire function that never hits 0, and this will also be an entire function (according to the Weierstrass factorization theorem). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than 0 an infinite number of times. Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times. Liouville's theorem is a special case of the following statement: Theorem: Assume ''M, R'' are positive constants and that ''n'' is a non-negative integer. An entire function ''f'' satisfying the inequality for all ''z'' with , is necessarily a polynomial, of degree at most ''n''.〔The converse is also true as for any polynomial of degree ''n'' the inequality holds for any |''z''| ≥ 1.〕 Similarly, an entire function ''f'' satisfying the inequality for all ''z'' with , is necessarily a polynomial, of degree at least ''n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Entire function」の詳細全文を読む スポンサード リンク
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