# at a point

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##### 1: 1.9 Calculus of a Complex Variable
###### Continuity
A function $f(z)$ is continuous at a point $z_{0}$ if $\lim\limits_{z\to z_{0}}f(z)=f(z_{0})$. … A function $f(z)$ is differentiable at a point $z$ if the following limit exists: … Conversely, if at a given point $(x,y)$ the partial derivatives $\ifrac{\partial u}{\partial x}$, $\ifrac{\partial u}{\partial y}$, $\ifrac{\partial v}{\partial x}$, and $\ifrac{\partial v}{\partial y}$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+iy$. … A function analytic at every point of $\mathbb{C}$ is said to be entire. …
##### 2: 10.2 Definitions
This solution of (10.2.1) is an analytic function of $z\in\mathbb{C}$, except for a branch point at $z=0$ when $\nu$ is not an integer. … Whether or not $\nu$ is an integer $Y_{\nu}\left(z\right)$ has a branch point at $z=0$. … Each solution has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. …
##### 3: 1.5 Calculus of Two or More Variables
###### §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
###### §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 $f\in C(a,b)$. … 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
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 $F((z_{0}-a)e^{2k\pi i}+a)$. …
##### 6: 22.19 Physical Applications Figure 22.19.1: Jacobi’s amplitude function am ⁡ ( x , k ) for 0 ≤ x ≤ 10 ⁢ π and k = 0.5 , 0.9999 , 1.0001 , 2 . …This corresponds to the pendulum being “upside down” at a point of unstable equilibrium. … Magnify
##### 7: 1.13 Differential Equations
A solution becomes unique, for example, when $w$ and $\ifrac{\mathrm{d}w}{\mathrm{d}z}$ are prescribed at a point in $D$. …
##### 8: 10.25 Definitions
It has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. …
##### 9: 15.6 Integral Representations
In (15.6.2) the point $\ifrac{1}{z}$ lies outside the integration contour, $t^{b-1}$ and $(t-1)^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\infty)$, and $(1-zt)^{a}=1$ at $t=0$. … In (15.6.5) the integration contour starts and terminates at a point $A$ on the real axis between $0$ and $1$. …
##### 10: 13.4 Integral Representations
The contour of integration starts and terminates at a point $\alpha$ on the real axis between $0$ and $1$. …