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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Cauchy–Hadamard theorem is a result in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
named after the French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s Augustin Louis Cauchy and
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a tea ...
, describing the radius of convergence of a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis.


Theorem for one complex variable

Consider the formal
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
in one complex variable ''z'' of the form f(z) = \sum_^ c_ (z-a)^ where a, c_n \in \Complex. Then the radius of convergence R of ''f'' at the point ''a'' is given by \frac = \limsup_ \left( , c_ , ^ \right) where denotes the
limit superior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
, the limit as approaches infinity of the
supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
of the sequence values after the ''n''th position. If the sequence values is unbounded so that the is ∞, then the power series does not converge near , while if the is 0 then the radius of convergence is ∞, meaning that the series converges on the entire plane.


Proof

Without loss of generality ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicat ...
assume that a=0. We will show first that the power series \sum_n c_n z^n converges for , z, , and then that it diverges for , z, >R. First suppose , z, . Let t=1/R not be 0 or \pm\infty. For any \varepsilon > 0, there exists only a finite number of n such that \sqrt \geq t+\varepsilon. Now , c_n, \leq (t+\varepsilon)^n for all but a finite number of c_n, so the series \sum_n c_n z^n converges if , z, < 1/(t+\varepsilon). This proves the first part. Conversely, for \varepsilon > 0, , c_n, \geq (t-\varepsilon)^n for infinitely many c_n, so if , z, =1/(t-\varepsilon) > R, we see that the series cannot converge because its ''n''th term does not tend to 0.


Theorem for several complex variables

Let \alpha be an ''n''-dimensional vector of natural numbers (\alpha = (\alpha_1, \cdots, \alpha_n) \in \N^n) with \, \alpha\, := \alpha_1 + \cdots + \alpha_n, then f(z) converges with radius of convergence \rho = (\rho_1, \cdots, \rho_n) \in \R^n, \rho^\alpha = \rho_1^ \cdots \rho_n^
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
\limsup_ \sqrt \alpha\, 1 of the multidimensional power series f(z) = \sum_c_\alpha(z-a)^\alpha := \sum_c_(z_1-a_1)^\cdots(z_n-a_n)^.


Proof

From Set z = a + t\rho (z_i = a_i + t\rho_i). Then :\sum_ c_\alpha (z - a)^\alpha = \sum_ c_\alpha \rho^\alpha t^ = \sum_ \left( \sum_ , c_\alpha, \rho^\alpha \right) t^\mu. This is a power series in one variable t which converges for , t, < 1 and diverges for , t, > 1. Therefore, by the Cauchy–Hadamard theorem for one variable :\limsup_ \sqrt mu= 1. Setting , c_m, \rho^m = \max_ , c_\alpha, \rho^\alpha gives us an estimate :, c_m, \rho^m \leq \sum_ , c_\alpha, \rho^\alpha \leq (\mu + 1)^n , c_m, \rho^m. Because \sqrt mu\to 1 as \mu \to \infty :\sqrt mu\leq \sqrt mu\leq \sqrt mu\implies \sqrt mu= \sqrt mu\qquad (\mu \to \infty). Therefore :\limsup_ \sqrt \alpha\, = \limsup_ \sqrt mu= 1.


Notes


External links

* {{DEFAULTSORT:Cauchy-Hadamard theorem Augustin-Louis Cauchy Series (mathematics) Theorems in complex analysis