# at a point

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## 1—10 of 103 matching pages

##### 1: 1.9 Calculus of a Complex Variable

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###### Continuity

►A function $f(z)$ is*continuous*at a point ${z}_{0}$ if $\underset{z\to {z}_{0}}{lim}f(z)=f({z}_{0})$. … ►A function $f(z)$ is*complex differentiable*at a point $z$ if the following limit exists: … ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … ►A function analytic at every point of $\u2102$ is said to be*entire*. …##### 2: 10.2 Definitions

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►This solution of (10.2.1) is an analytic function of $z\in \u2102$, except for a branch point at
$z=0$ when $\nu $ is not an integer.
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►Whether or not $\nu $ is an integer ${Y}_{\nu}\left(z\right)$ has a branch point at
$z=0$.
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►Each solution has a branch point at
$z=0$ for all $\nu \in \u2102$.
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##### 3: 1.5 Calculus of Two or More Variables

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###### §1.5(i) Partial Derivatives

►A function $f(x,y)$ is*continuous at a point*$(a,b)$ if … ►A function is*continuous on a point set*$D$ if it is continuous at all points of $D$. …##### 4: 1.4 Calculus of One Variable

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###### §1.4(ii) Continuity

… ►If $f(x)$ is continuous at each point $c\in (a,b)$, then $f(x)$ is*continuous on the interval*$(a,b)$ and we write $f\in C(a,b)$. … ►For historical reasons, $w(x)$ is also sometimes referred to as a*density*, as, for example, the mass per unit length at point $x$, see Shohat and Tamarkin (1970, p vii). … ►The overall maximum (minimum) of $f(x)$ on $[a,b]$ will either be at a local maximum (minimum) or at one of the end points $a$ or $b$. … ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …##### 5: 1.10 Functions of a Complex Variable

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►A function whose only singularities, other than the point at infinity, are poles is called a

*meromorphic function*. … ►Functions which have more than one value at a given point $z$ are called*multivalued*(or*many-valued*) functions. …If we can assign a unique value $f(z)$ to $F(z)$ at each point of $D$, and $f(z)$ is analytic on $D$, then $f(z)$ is a*branch*of $F(z)$. … ►(b) By specifying the value of $F(z)$ at a point ${z}_{0}$ (not a branch point), and requiring $F(z)$ to be continuous on any path that begins at ${z}_{0}$ and does not pass through any branch points or other singularities of $F(z)$. ►If the path circles a branch point at $z=a$, $k$ times in the positive sense, and returns to ${z}_{0}$ without encircling any other branch point, then its value is denoted conventionally as $F(({z}_{0}-a){\mathrm{e}}^{2k\pi \mathrm{i}}+a)$. …##### 6: 22.19 Physical Applications

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##### 7: 10.25 Definitions

##### 8: 15.6 Integral Representations

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►In (15.6.2) the point
$1/z$ lies outside the integration contour, ${t}^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty})$, and ${(1-zt)}^{a}=1$
at
$t=0$.
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►In (15.6.5) the integration contour starts and terminates at a point
$A$ on the real axis between $0$ and $1$.
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##### 9: 13.4 Integral Representations

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►The contour of integration starts and terminates at a point
$\alpha $ on the real axis between $0$ and $1$.
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##### 10: 1.13 Differential Equations

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►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$.
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►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda $; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called

*nodes*, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …