By Moore C.N.
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Extra resources for On the Application of Borels Method to the Summation of Fouriers Series
In particular, I0 I1, or fzdz fzdz. C1 C2 This is a big deal. We have shown that if C 1 and C 2 are closed curves in a region D that are homotopic in D, and f is analytic on D, then fzdz fzdz. 3 fzdz 0. C In court testimony, one is admonished to tell the truth, the whole truth, and nothing but the truth. Well, so far in this chapter, we have told the truth and nothing but the truth, but we have not quite told the whole truth. We assumed all sorts of continuous derivatives in the preceding discussion.
Explain. 16. Find the principal value of a) i i . b) 1 i 4i 17. a)Find all values of |i i |. 1. Introduction. If : D C is simply a function on a real interval D , , then the integral tdt is, of course, simply an ordered pair of everyday 3 rd grade calculus integrals: tdt xtdt i ytdt, where t xt iyt. Thus, for example, 1 t 2 1 it 3 dt 0 4 i. 4 3 Nothing really new here. The excitement begins when we consider the idea of an integral of an honest-to-goodness complex function f : D C, where D is a subset of the complex plane.
5 5 i. 6 Thus, fzdz fzdz fzdz C3 C 31 C 32 5 i. 6 Suppose there is a number M so that |fz| M for all zC. Then fzdz ft tdt C |ft t|dt M | t|dt ML, where L | t|dt is the length of C. Exercises 1. Evaluate the integral z dz, where C is the parabola y x 2 from 0 to 1 i. C 2. Evaluate 1 z dz, where C is the circle of radius 2 centered at 0 oriented C counterclockwise. 4. Evaluate fzdz, where C is the curve y x 3 from 1 i to 1 i , and C 1 fz for y 0 4y for y 0 .