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1: 1.9 Calculus of a Complex Variable
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Continuity
►A function is continuous at a point if . … ►A function is complex differentiable at a point if the following limit exists: … ►Conversely, if at a given point the partial derivatives , , , and exist, are continuous, and satisfy (1.9.25), then is differentiable at . … ►A function analytic at every point of is said to be entire. …2: 10.2 Definitions
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►This solution of (10.2.1) is an analytic function of , except for a branch point at
when is not an integer.
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►Whether or not is an integer has a branch point at
.
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►Each solution has a branch point at
for all .
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3: 1.5 Calculus of Two or More Variables
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§1.5(i) Partial Derivatives
►A function is continuous at a point if … ►A function is continuous on a point set if it is continuous at all points of . …4: 1.4 Calculus of One Variable
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§1.4(ii) Continuity
… ►If is continuous at each point , then is continuous on the interval and we write . … ►For historical reasons, is also sometimes referred to as a density, as, for example, the mass per unit length at point , see Shohat and Tamarkin (1970, p vii). … ►The overall maximum (minimum) of on will either be at a local maximum (minimum) or at one of the end points or . … ►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.
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►Functions which have more than one value at a given point
are called multivalued (or many-valued) functions.
…If we can assign a unique value to
at each point of , and is analytic on , then is a branch of .
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►(b) By specifying the value of
at a point
(not a branch point), and requiring to be continuous on any path that begins at
and does not pass through any branch points or other singularities of .
►If the path circles a branch point at
, times in the positive sense, and returns to without encircling any other branch point, then its value is denoted conventionally as .
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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
lies outside the integration contour, and assume their principal values where the contour cuts the interval , and
at
.
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►In (15.6.5) the integration contour starts and terminates at a point
on the real axis between and .
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9: 13.4 Integral Representations
10: 1.13 Differential Equations
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►A solution becomes unique, for example, when and are prescribed at a point in .
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►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, ; (ii) the corresponding (real) eigenfunctions, and , have the same number of zeros, also called nodes, for as for ; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points.
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