divergence theorem

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1: 1.6 Vectors and Vector-Valued Functions

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Gauss’s (or Divergence) Theorem

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2: 15.2 Definitions and Analytical Properties

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Diverges when $\mathrm{\Re }\left(c-a-b\right)\le -1$.

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3: 2.1 Definitions and Elementary Properties

… ►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $𝐒$ and $f(z)=O\left(z^{\nu }\right)$ as $z\to \mathrm{\infty }$ in $𝐒$, $\nu$ being real, then $f^{\prime }(z)=O\left(z^{\nu -1}\right)$ as $z\to \mathrm{\infty }$ in any closed sector properly interior to $𝐒$ and with the same vertex (Ritt’s theorem). This result also holds with both $O$’s replaced by $o$’s. … ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, …

4: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

… ►If the limit exists, then the double series is convergent; otherwise it is divergent. … ►

Dominated Convergence Theorem

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5: 16.2 Definition and Analytic Properties

… ►On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\mathrm{\Re }{\gamma }_q>0$, convergent except at $z=1$ if , and divergent if $\mathrm{\Re }{\gamma }_q\le -1$, where … ►In general the series (16.2.1) diverges for all nonzero values of $z$. …

6: 2.7 Differential Equations

… ►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda }}_j$ is zero) they diverge. … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: ►

Liouville–Green Approximation Theorem

… ►We cannot take $f=x$ and $g=\ln x$ because $\int g{f}^{-1/2}dx$ would diverge as $x\to +\mathrm{\infty }$. …