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As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere [d] 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.
Similarly, if an entire function has a pole of order at —that is, it grows in magnitude comparably to in some neighborhood of —then is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if | f ( z ) | ≤ M | z | n {\displaystyle |f(z)|\leq M|z|^{n}} for | z | {\displaystyle |z|} sufficiently large ...
Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions , are holomorphic over the entire complex plane, making them entire functions, while rational functions /, where p and q are polynomials, are holomorphic on domains ...
Define the Hadamard canonical factors ():= = / Entire functions of finite order have Hadamard's canonical representation: [1] = = (/) where are those roots of that are not zero (), is the order of the zero of at = (the case = being taken to mean ()), a polynomial (whose degree we shall call ), and is the smallest non-negative integer such that the series = | | + converges.
A result of the Great Picard Theorem is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception. The "single exception" is needed in both theorems, as demonstrated here: e z is an entire non-constant function that is never 0,
Image credits: BaronVonBroccoli In the battle of aesthetics vs. function, visually pleasing appearance seems to be winning. As the theory of the aesthetic-usability effect suggests, users tend to ...
Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet. The complex conjugate function z → z * is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from R ...
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