at a point

SearchAdvancedHelp

(0.010 seconds)

1—10 of 103 matching pages

1: 1.9 Calculus of a Complex Variable

… ►

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 $ℂ$ is said to be entire. …

2: 10.2 Definitions

… ►This solution of (10.2.1) is an analytic function of $z\in ℂ$, 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 $\nu \in ℂ$. …

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 $f\in C(a,b)$. … ►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

… ►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

… ►

►►

Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ 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: 10.25 Definitions

… ►It has a branch point at $z=0$ for all $\nu \in ℂ$. …

8: 15.6 Integral Representations

… ►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 $t=0$. … ►In (15.6.5) the integration contour starts and terminates at a point $A$ on the real axis between $0$ and $1$. …

9: 13.4 Integral Representations

… ►The contour of integration starts and terminates at a point $\alpha$ on the real axis between $0$ and $1$. …

10: 1.13 Differential Equations

… ►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ►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. …