continuous function
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1: 1.4 Calculus of One Variable
2: 4.12 Generalized Logarithms and Exponentials
3: 2.8 Differential Equations with a Parameter
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►in which ranges over a bounded or unbounded interval or domain , and is or analytic on .
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►Again, and is on .
Corresponding to each positive integer there are solutions , , that are on , and as
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►Also, is on , and .
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►In the former, corresponding to any positive integer there are solutions , , that are on , and as
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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►and functions
, assumed real for the moment.
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►For , has the eigenfunction expansion, following directly from (1.18.17)–(1.18.19),
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►For , has the eigenfunction expansion, analogous to that of (1.18.33),
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►More generally, for , , see (1.4.24),
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►, ) of which is moreover in .
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5: 3.5 Quadrature
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►where , , and .
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►If in addition is periodic, , and the integral is taken over a period, then
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►Let and .
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►If , then the remainder in (3.5.2) can be expanded in the form
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►For
functions Gauss quadrature can be very efficient.
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6: 6.16 Mathematical Applications
7: 1.13 Differential Equations
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and belong to domains and respectively, the coefficients and are continuous functions of both variables, and for each fixed (fixed ) the two functions are analytic in (in ).
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►As the interval is mapped, one-to-one, onto by the above definition of , the integrand being positive, the inverse of this same transformation allows to be calculated from in (1.13.31), and .
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8: 1.17 Integral and Series Representations of the Dirac Delta
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1.17.2
,
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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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1.17.3
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1.17.6
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1.17.9
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9: 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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10: 1.8 Fourier Series
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►Let be an absolutely integrable function of period , and continuous except at a finite number of points in any bounded interval.
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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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