Introduction to the Theory of Fourier's Series and Integrals Volume 1

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 edition. Excerpt: ...the same is true if (x--ayf(x) tends to +00, or to--00, as cc-- + 0. We shall speak of this test as the-test for the infinite integral I f(x)dx, when x = a is a point of infinite discontinuity. It is J a clear that in applying this test we are simply asking ourselves the order of the infinity that occurs in the integrand. The results can he readily adapted to the case when the upper limit b is a point of infinite discontinuity. Also, it is easy to show that VI. If px) is bounded and integrable in (a, b), and f fs(x)dx converges absolutely, then f p(x)ys(x)dx is absolutely J a Jit convergent. (Cf. 56, I.) The tests given in III.-VI. will cover most of the cases which we shall meet. But it would not be difficult to develop in detail the results which correspond to the other tests obtained for the poo convergence of the infinite integral 1 f(x) dx. J a n No special discussion is required for the integral I f(x) dx, when a certain number of points of infinite discontinuity occur ft" Ex. 4. Show that / log sin xdx converges and is equal to-At log 2. The only infinity is at x=0, and the convergence of the integral follows from the fi-test. Further, / & / log %mxdx=% log sin 2.v dx Jo Jo =;rlog2 + 2/ log sin x dx + 2 / logcosA-ofo; Jo JO = 7r log 2 + 41 log sin x dx. Jo But Jo log sin xdx = 2 j logsin.rcfo, '. Therefore o log sin xdx =-- 71-log 2. From this result it is easy to show that the convergent integrals I log (1--cos x) dx and / log (1 + cos x) dx Jo Jo are equal to-Tt log 2. Ex. 5. Discuss the convergence or divergence of the Gamma Function integral J e'xnldx. (i) Let 1. Then the integrand is bounded in 0.i'- a, where a is arbitrary, and we need only consider the convergence of / e'xnl dx. Ja The /x-test of ...show more

Product details

• Paperback | 64 pages
• 189 x 246 x 3mm | 132g
• Miami Fl, United States
• English
• black & white illustrations
• 1236642228
• 9781236642226