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11: 28.2 Definitions and Basic Properties

… ►Furthermore, a solution $w$ with given initial constant values of $w$ and $w^{\prime }$ at a point $z_0$ is an entire function of the three variables $z$, $a$, and $q$. …

12: 2.4 Contour Integrals

… ►Cases in which $p^{\prime }(t_0)\ne 0$ are usually handled by deforming the integration path in such a way that the minimum of $\mathrm{\Re }\left(zp(t)\right)$ is attained at a saddle point or at an endpoint. …

13: 25.12 Polylogarithms

… ►In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. …

14: 10.61 Definitions and Basic Properties

… ►In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $x{\mathrm{e}}^{\pm \pi \mathrm{i}}$ are complex. …

15: 32.11 Asymptotic Approximations for Real Variables

… ►If $|k|>1$, then $w_k(x)$ has a pole at a finite point $x=c_0$, dependent on $k$, and …

16: 8.19 Generalized Exponential Integral

… ►Unless $p$ is a nonpositive integer, $E_p\left(z\right)$ has a branch point at $z=0$. …

17: Mathematical Introduction

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$ℂ$	complex plane (excluding infinity).
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${f(z)|}_C=0$	$f(z)$ is continuous at all points of a simple closed contour $C$ in $ℂ$.
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18: 1.8 Fourier Series

… ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. …at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. …

19: 2.11 Remainder Terms; Stokes Phenomenon

… ►When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation. … ►For large $\rho$ the integrand has a saddle point at $t={\mathrm{e}}^{-\mathrm{i}\theta }$. …

20: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. …