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  • Cauchy-Hadamard Test – Mathematics Notes – For W.B.C.S. Examination.
    Posted on October 21st, 2019 in Mathematics
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    Cauchy-Hadamard Test – Mathematics Notes – For W.B.C.S. Examination.

    কাউচি-হাডামার্ড পরীক্ষা – গণিতের নোট – WBCS পরীক্ষা।

    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,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.Consider the formal power series in one complex variable z of the form.Continue Reading Cauchy-Hadamard Test – Mathematics Notes – For W.B.C.S. Examination.

    {\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}}

    where {\displaystyle a,c_{n}\in \mathbb {C} .}

    Then the radius of convergence {\displaystyle R} of ƒ at the point a is given by

    {\displaystyle {\frac {1}{R}}=\limsup _{n\to \infty }{\big (}|c_{n}|^{1/n}{\big )}}

    where lim sup denotes the limit superior, the limit as n approaches infinity of the supremum of the sequence values after the nth position. If the sequence values are unbounded so that the lim sup is ∞, then the power series does not converge near a, while if the lim sup is 0 then the radius of convergence is ∞, meaning that the series converges on the entire plane.

    Proof of the theorem

    Without loss of generality assume that {\displaystyle a=0}. We will show first that the power series {\displaystyle \sum c_{n}z^{n}} converges for {\displaystyle |z|<R}<img class=”mwe-math-fallback-image-inline” src=”https://wikimedia.org/api/rest_v1/media/math/render/svg/60c24dbdb8afa15bfa5c563fa13d29b8fa68899d” alt=”|z|, and then that it diverges for {\displaystyle |z|>R}R” aria-hidden=”true”>.

    First suppose {\displaystyle |z|<R}<img class=”mwe-math-fallback-image-inline” src=”https://wikimedia.org/api/rest_v1/media/math/render/svg/60c24dbdb8afa15bfa5c563fa13d29b8fa68899d” alt=”|z|. Let {\displaystyle t=1/R} not be zero or ±infinity. For any {\displaystyle \varepsilon >0}0″ aria-hidden=”true”>, there exists only a finite number of {\displaystyle n} such that {\displaystyle {\sqrt[{n}]{|c_{n}|}}\geq t+\varepsilon }. Now {\displaystyle |c_{n}|\leq (t+\varepsilon )^{n}} for all but a finite number of {\displaystyle c_{n}}, so the series {\displaystyle \sum c_{n}z^{n}} converges if {\displaystyle |z|<1/(t+\varepsilon )}<img class=”mwe-math-fallback-image-inline” src=”https://wikimedia.org/api/rest_v1/media/math/render/svg/af16887fa22189a614cc6b70d928d91fd078d5df” alt=”{\displaystyle |z|. This proves the first part.

    Conversely, for {\displaystyle \varepsilon >0}0″ aria-hidden=”true”>{\displaystyle |c_{n}|\geq (t-\varepsilon )^{n}} for infinitely many {\displaystyle c_{n}}, so if {\displaystyle |z|=1/(t-\varepsilon )>R}R}” aria-hidden=”true”>, we see that the series cannot converge because its nth term does not tend to 0.

    Several complex variables

    Statement of the theorem

    Let {\displaystyle \alpha } be a multi-index (a n-tuple of integers) with {\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}}, then {\displaystyle f(x)} converges with radius of convergence {\displaystyle \rho } (which is also a multi-index) if and only if

    {\displaystyle \lim _{|\alpha |\to \infty }{\sqrt[{|\alpha |}]{|c_{\alpha }|\rho ^{\alpha }}}=1}

    to the multidimensional power series

    {\displaystyle \sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }:=\sum _{\alpha _{1}\geq 0,\ldots ,\alpha _{n}\geq 0}c_{\alpha _{1},\ldots ,\alpha _{n}}(z_{1}-a_{1})^{\alpha _{1}}\cdots (z_{n}-a_{n})^{\alpha _{n}}}
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