piecewise continuous functions
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11—15 of 15 matching pages
11: 1.17 Integral and Series Representations of the Dirac Delta
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►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that
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►for all functions
that are continuous when , and for each , converges absolutely for all sufficiently large values of .
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►More generally, assume is piecewise continuous (§1.4(ii)) when for any finite positive real value of , and for each , converges absolutely for all sufficiently large values of .
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►provided that is continuous when , and for each , converges absolutely for all sufficiently large values of (as in the case of (1.17.6)).
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►provided that is continuous and of period ; see §1.8(ii).
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12: 1.6 Vectors and Vector-Valued Functions
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►The path integral of a continuous function
is
…when is continuous, and
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►The curve is piecewise differentiable if is piecewise differentiable.
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►The integral of a continuous function
over a surface is
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13: About Color Map
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►Surface visualizations in the DLMF represent functions of the form by the height or the magnitude, , for complex functions, over the plane.
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►By painting the surfaces with a color that encodes the phase, , both the magnitude and phase of complex valued functions can be displayed.
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Continuous Phase Mapping
►For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. … ►We therefore use a piecewise linear mapping as illustrated below, that takes phase to red, to yellow, to cyan and to blue. …14: 1.9 Calculus of a Complex Variable
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