About the Project
NIST

at a point

AdvancedHelp

(0.004 seconds)

1—10 of 98 matching pages

1: 1.9 Calculus of a Complex Variable
Continuity
A function f ( z ) is continuous at a point z 0 if lim z 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 u / x , u / y , v / x , and v / y exist, are continuous, and satisfy (1.9.25), then f ( z ) is differentiable at z = x + i y . … A function analytic at every point of is said to be entire. …
2: 10.2 Definitions
This solution of (10.2.1) is an analytic function of z , except for a branch point at z = 0 when ν is not an integer. … Whether or not ν is an integer Y ν ( z ) has a branch point at z = 0 . … Each solution has a branch point at z = 0 for all ν . …
3: 1.4 Calculus of One Variable
§1.4(ii) Continuity
If f ( x ) is continuous at each point c ( a , b ) , then f ( x ) is continuous on the interval ( a , b ) and we write f 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. …
4: 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 2 k π i + a ) . …
5: 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 . …
6: 22.19 Physical Applications
See accompanying text
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 d w / d z are prescribed at a point in D . …
8: 10.25 Definitions
It has a branch point at z = 0 for all ν . …
9: 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 , ) , and ( 1 - z t ) 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 α on the real axis between 0 and 1 . …