Microcódigo

For the function giving the whole part of a number, to see Left whole .

In Analyze complexes, a function is known as whole if it is defined on all the plane complex and is holomorphic in any point.

Examples

• Of the typical examples of whole functions is the polynomial functions, the exponential function, and the sums, the products and the made up ones of those.

• the goniometrical functions and the hyperbolic functions are also whole functions, but they are simple variations of the exponential function in consequence of the formulas of Euler.

• Any series whole whose ray of convergence is infinite defines a whole function. Reciprocally, any whole function can be represented by a whole series whose ray of convergence is infinite.

• Neither the function natural Logarithme nor the function square Racine are whole functions.

Principal results

• If one poses $f\; (S)\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; \{a\_n\; (s-z)\; ^n\}$, and if M (R) indicates the maximum of the function on the disc of center Z and ray R, one has the priceless inequalities of Cauchy

$|a\_n|\; \backslash \; the\; \backslash \; frac\; \{M\; (R)\}\{R^n\}$

• the theory of Cauchy shows that the integral along a closed loop of a whole function is null.

• One with the formula of Cauchy

$F\; (Z)\; =\; \backslash \; frac1\; \{2\; \backslash \; pi\; I\}\; \backslash \; int\_\; \backslash \; gamma\; \{\backslash \; frac\; \{F\; (S)\}\{s-z\}ds\}$ and, by developing fraction 1 (s-z) in whole series one from of deduced that $a\_n=\; \backslash \; frac\; \{f^\; \{(N)\}(Z)\}\{N!\}=\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\; I\}\; \backslash \; int\_\; \backslash \; gamma\; \{\backslash \; frac\; \{F\; (S)\}\{(s-z)\; ^\; \{n+1\}\}\; ds\}$ where in both cases $\backslash \; gamma$ is a loop closed without loop containing Z.

• an important result on the whole functions is the theorem of Liouville: if a whole function is limited, then it is constant.

• That can be used to provide an elegant demonstration, by the absurdity, of the Théorème of Alembert-Gauss.

• the small theorem of Picardy reinforces the theorem of Liouville considerably.

Notice

• a function which is definite and holomorphic on all the complex level except on a whole of isolated poles is known as méromorphe.

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