Cauchy–Hadamard theorem explained

In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy,^[1] but remained relatively unknown until Hadamard rediscovered it.^[2] Hadamard's first publication of this result was in 1888;^[3] he also included it as part of his 1892 Ph.D. thesis.^[4]

Theorem for one complex variable

Consider the formal power series in one complex variable z of the form$$f(z)\; =\; \backslash sum\_^\; c\_\; (z-a)^$$where

a, c _{n} \in \Complex. R of f at the point a is given by$$\backslash frac\; =\; \backslash limsup\_\; \backslash left(|\; c\_\; |^\; \backslash right)$$where denotes the limit superior, the limit as approaches infinity of the supremum of the sequence values after the nth position. If the sequence values are 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 assume that

a =0 . We will show first that the power series $\backslash sum\_n\; c\_n\; z^n$ converges for |z|<R , and then that it diverges for |z|>R .

First suppose

|z|<R . Let t =1 /R not be 0 or \pm inf ty. For any \varepsilon > 0 , there exists only a finite number of n such that $\backslash sqrt[n]$ \geq t+\varepsilon. Now |c _{n|} \leq (t + \varepsilon) ^{n} for all but a finite number of c _{n} , so the series $\backslash sum\_n\; c\_n\; z^n$ 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 nth term does not tend to 0.^[5]

Theorem for several complex variables

Let

\alpha be an n-dimensional vector of natural numbers (\alpha = (\alpha _{1,} … , \alpha _{n)} \in \N ^{n} ) with ||\alpha|| = \alpha _{1} + … + \alpha _{n} , then f(x) converges with radius of convergence \rho = (\rho _{1,} … , \rho _{n)} \in \R ^{n} with \rho ^{\alpha} = \rho ^{\alpha _{1}} _{1} … \rho ^{\alpha _{n}} _{n} if and only if$$\backslash limsup\_$$

\alpha

\to\infty

\sqrt[||\alpha||]

\rho^\alpha

=1to the multidimensional power series$$\backslash sum\_c\_\backslash alpha(z-a)^\backslash alpha\; :=\; \backslash sum\_c\_(z\_1-a\_1)^\backslash cdots(z\_n-a\_n)^$$

Proof

From

Set

z = a + t\rho (z _{i} = a _{i} + t\rho _{i)} , then\sum _{\alpha} c _{\alpha} (z - a) ^{\alpha} = \sum _{\alpha} c _{\alpha} \rho ^{\alpha} t ^{||\alpha||} = \sum _{\mu} \left( \sum _{||\alpha||} |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 _{\mu} \sqrt[\mu]{\sum _{||\alpha||} |c _{\alpha|} \rho ^{\alpha}} = 1

Setting

|c _{m|} \rho ^{m} = max _{||\alpha||} |c _{\alpha|} \rho ^{\alpha} gives us an estimate|c _{m|} \rho ^{m} \leq \sum _{||\alpha||} |c _{\alpha|} \rho ^{\alpha} \leq (\mu + 1 ) ^{n} |c _{m|} \rho ^{m}

Because

\sqrt[\mu]{(\mu + 1 ) ^{n}} \to 1 as \mu \to inf ty \sqrt[\mu]{|c _{m|} \rho ^{m}} \leq \sqrt[\mu]{\sum _{||\alpha||} |c _{\alpha|} \rho ^{\alpha}} \leq \sqrt[\mu]{|c _{m|} \rho ^{m}} \implies \sqrt[\mu]{\sum _{||\alpha||} |c _{\alpha|} \rho ^{\alpha}} = \sqrt[\mu]{|c _{m|} \rho ^{m}} (\mu \to inf ty)

Therefore

\limsup _{||\alpha||\toinfty} \sqrt[||\alpha||]{|c ^{\alpha}} _{\alpha|\rho} = \limsup _{\mu} \sqrt[\mu]{|c _{m|} \rho ^{m}} = 1

Notes and References

1. .
2. . Translated from the Italian by Warren Van Egmond.
3. .
4. . Also in Thèses présentées à la faculté des sciences de Paris pour obtenir le grade de docteur ès sciences mathématiques, Paris: Gauthier-Villars et fils, 1892.
5. Graduate Texts in Mathematics