►A function is continuousatapoint
if .
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►A function is complex differentiableatapoint
if the following limit exists:
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►Conversely, if ata given point
the partial derivatives , , , and exist, are continuous, and satisfy (1.9.25), then is differentiable at
.
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►A function analytic at every point of is said to be entire.
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►This solution of (10.2.1) is an analytic function of , except for a branch pointat
when is not an integer.
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►Whether or not is an integer has a branch pointat
.
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►Each solution has a branch pointat
for all .
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►If is continuous at each point
, then is continuous on the interval
and we write .
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►For historical reasons, is also sometimes referred to as adensity, as, for example, the mass per unit length atpoint
, see Shohat and Tamarkin (1970, p vii).
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►The overall maximum (minimum) of on will either be ata local maximum (minimum) or at one of the end points
or .
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►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.
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►A function whose only singularities, other than the pointat infinity, are poles is called ameromorphic function.
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►Functions which have more than one value ata 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 abranch of .
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►(b) By specifying the value of
atapoint
(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 pointat
, 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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►►►Figure 22.19.1: Jacobi’s amplitude function for and .
…This corresponds to the pendulum being “upside down” atapoint of unstable equilibrium.
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Magnify
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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 atapoint
on the real axis between and .
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►A solution becomes unique, for example, when and are prescribed atapoint 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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