piecewise differentiable
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9 matching pages
1: 1.6 Vectors and Vector-Valued Functions
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►The curve is piecewise differentiable if is piecewise differentiable.
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►Sufficient conditions for this result to hold are that and are continuously differentiable on , and is piecewise differentiable.
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2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
3: 1.5 Calculus of Two or More Variables
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►A function is piecewise continuous on , where and are intervals, if it is piecewise continuous in for each and piecewise continuous in for each .
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►If is times continuously differentiable, then
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►Sufficient conditions for the limit to exist are that is continuous, or piecewise continuous, on .
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►Moreover, if are finite or infinite constants and is piecewise continuous on the set , then
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4: 1.9 Calculus of a Complex Variable
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Differentiation
►A function is complex differentiable at a point if the following limit exists: … ►A function is said to be analytic (holomorphic) at if it is complex differentiable in a neighborhood of . … ►An arc is given by , , where and are continuously differentiable. If and are continuous and and are piecewise continuous, then defines a contour. …5: 10.43 Integrals
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On the interval , is continuously differentiable and each of and is absolutely integrable.
is piecewise continuous and of bounded variation on every compact interval in , and each of the following integrals
6: 1.8 Fourier Series
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►If is of period , and is piecewise continuous, then
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►For
piecewise continuous on and real ,
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►If and are the Fourier coefficients of a piecewise continuous function on , then
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►If a function is periodic, with period , then the series obtained by differentiating the Fourier series for term by term converges at every point to .
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►Suppose that is twice continuously differentiable and and are integrable over .
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7: 1.4 Calculus of One Variable
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►For an example, see Figure 1.4.1
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►When this limit exists is differentiable at .
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Mean Value Theorem
… ►Continuity, or piecewise continuity, of on is sufficient for the limit to exist. …8: 3.7 Ordinary Differential Equations
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►Let be a finite or infinite interval and be a real-valued continuous (or piecewise continuous) function on the closure of .
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►If is on the closure of , then the discretized form (3.7.13) of the differential equation can be used.
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9: 2.3 Integrals of a Real Variable
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►converges for all sufficiently large , and is infinitely differentiable in a neighborhood of the origin.
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►If, in addition, is infinitely differentiable on and
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►assume and are finite, and is infinitely differentiable on .
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►In addition to (2.3.7) assume that and are piecewise continuous (§1.4(ii)) on , and
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On , and are infinitely differentiable and .