A Fourier Analysis And Its Applications by Reinhard Diestel PDF

By Reinhard Diestel

ISBN-10: 0387950141

ISBN-13: 9780387950143

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Now let s be a real (or complex, if you like) number. 1) 0 exists (with a finite value), we say that it is the Laplace transform of f , evaluated at the point s. We shall write, interchangeably, f (s) or L[f ](s). In applications, one also often uses the notation F (s) (capital letter for the transform of the corresponding lower-case letter). 1. Let f (t) = eat , t ≥ 0. Then, ∞ 0 ∞ f (t) e−st dt = eat−st dt = 0 e(a−s)t a−s ∞ = t=0 1 , s−a provided that a − s < 0 so that the evaluation at infinity yields zero.

Ak−1 k=1 This is a sum of finitely many terms, and by the preceding case each of these terms tends to zero as λ → ∞. Thus the assertion is true also for this f. Let now f be an arbitrary function that is Riemann integrable on I = [a, b]. Let ε be an arbitrary positive number. By the definition of the Riemann integral, there exists a piecewise constant function g such that b |f (u) − g(u)| du < a ε . ) Then, b b a b (f (u) − g(u)) sin λu du + f (u) sin λu du = a g(u) sin λu du a b ≤ b |f (u) − g(u)|| sin λu| du + a b ≤ g(u) sin λu du a b |f (u) − g(u)| du + a g(u) sin λu du .

This can be expressed by writing f (t) = f (t)H(t), where H is the Heaviside function. Now let s be a real (or complex, if you like) number. 1) 0 exists (with a finite value), we say that it is the Laplace transform of f , evaluated at the point s. We shall write, interchangeably, f (s) or L[f ](s). In applications, one also often uses the notation F (s) (capital letter for the transform of the corresponding lower-case letter). 1. Let f (t) = eat , t ≥ 0. Then, ∞ 0 ∞ f (t) e−st dt = eat−st dt = 0 e(a−s)t a−s ∞ = t=0 1 , s−a provided that a − s < 0 so that the evaluation at infinity yields zero.

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A Fourier Analysis And Its Applications by Reinhard Diestel


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