…
►in which ranges over a bounded or unbounded interval or domain , and is or analytic on .
…
►Again, and is on .
Corresponding to each positive integer there are solutions , , that are on , and as
…
►Also, is on , and .
…
►In the former, corresponding to any positive integer there are solutions , , that are on , and as
…
…
►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),
…
►, ) of which is moreover in .
…
…
►where , , and .
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►If in addition is periodic, , and the integral is taken over a period, then
…
►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.
…
…
►
and belong to domains and respectively, the coefficients and are continuousfunctions of both variables, and for each fixed (fixed ) the two functions are analytic in (in ).
…
►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 .
…
…
►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.
…
…
►Let be an absolutely integrable function of period , and continuous except at a finite number of points in any bounded interval.
…
►If and are the Fourier coefficients of a piecewise continuousfunction
on , then
…
►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 .
…