By B K Dutta
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W. H. Freeman and Company, New York. , and Toupin, R. (1960), The Classical Field Theories. In Handbuch der Physik (S. ), Band III/l, Springer-Verlag, Berlin. Wrede, R. C. (1972), Introduction to Vector and Tensor Analysis, Dover, New York. 1 Fourier Series Let f (x) be a continuous, integrable function deﬁned on the interval [−c, c]. , f (x) = (a0 /2) + ∞ [ak cos(kπ x/c) + bk sin(kπ x/c)]. 1) k=1 The coefﬁcients, ak and bk, indexed by the integers k, can be identiﬁed as follows. 1) by cos(nπ x/c), n being an integer, and integrate over [−c, c] to obtain c c f (x) cos(nπ x/c) dx = −c −c + (a0 /2) cos(nπ x/c) dx c ∞ −c k=1 [ak cos(kπ x/c) cos(nπ x/c) + bk sin(kπ x/c) cos(nπ x/c)] dx.
These are clearly orthogonal on the square deﬁned by −π ≤ x ≤ π and −π ≤ y ≤ π. For reference we note that the norms are 1 = 2π, cos(mx) = sin(mx) = √ 2π , cos(mx) cos(ny) = sin(mx) sin(ny) = cos(mx) sin(ny) = π. 19) f (x, y) cos(mx) sin(ny) dx dy, f (x, y) sin(mx) sin(ny) dx dy, for m, n = 1, 2, . . For the cases where either m = 0 or n = 0, we have Am0 = R = 1 2π 2 f (x, y) cos(mx) dx dy cos(mx) 2 f (x, y) cos(mx) dx dy, (m = 1, 2, . 3. Integral Transforms 39 A0n = R = 1 2π 2 Bm0 = R = 1 2π 2 B0n = R = 1 2π 2 f (x, y) cos(ny) dx dy cos(ny) 2 f (x, y) cos(ny) dx dy, (n = 1, 2, .
0 Now let η ≡ αx, and write ∞ ∞ K(αx)α s−1 dα = x −s 0 K(η)ηs−1 dη = x −s K(s). 33) 0 = K(s)F(1 − s). 35) ∞ I f (α) dα 0 x s−1 K(αx) dx. 3. 38) as desired. 40) Fc (α) cos(αx) dα. 41) 0 with its inverse f (x) = ∞ 2/π 0 As another example, let K(αx) be deﬁned as K(αx) = 2/π sin(αx). 43) Fs (α) sin(αx) dα. 47) where K(s) and H(s) are the Mellin transforms of K(x) and H(x), respectively. 48) f (x) = π 0 0 or, more generally, f (x) = 1 π ∞ −∞ ∞ dα −∞ f (η) cos(αη) cos(αx) dη. 49) Before exploring this, it is necessary to establish some integrability properties of f (x).