By Reinhard Diestel
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Extra info for A Fourier Analysis And Its Applications
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.
These methods can be eﬃcient if the terms in the series have diﬀerent signs or are complex numbers. A positive divergent series cannot be summed to anything but +∞, no matter how many means you try. , the series k=1 ak , where a3k+1 = 1, a3k+2 = 0 and a3k+3 = −1. Compute the Ces` aro means σn and show that the series has the Ces` aro sum 23 . 6 and the previous exercise can be generalized as follows. , that there is a positive integer p such that sn+p = sn for all n. Then the series is summable (C, 1) to the sum σ = (s1 + s2 + · · · + sp )/p.
And the theorem can be applied. The number σ0 is a reasonably exact measure of the rate of growth of the function f . In what follows we shall sometimes use the notation σ0 or σ0 (f ) for this measure. As a consequence of the theorem we now know that a large set of common functions do have Laplace transforms. , polynomials, trigonometric functions such as sin and cos and ordinary exponential functions; also sums and products of such functions. If you have studied simple diﬀerential equations you may recall that these functions are precisely the possible solutions of homogeneous linear diﬀerential equations with constant coeﬃcients, such as, for example, y (v) + 4y (iv) − 8y + 15y − 24y = 0.