ch6 - Chapter Six More Integration 6.1. Cauchys Integral...

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Chapter Six More Integration 6.1. Cauchy’s Integral Formula. Suppose f is analytic in a region containing a simple closed contour C with the usual positive orientation and its inside , and suppose z 0 is inside C . Then it turns out that f z 0 1 2 i C f z z z 0 dz . This is the famous Cauchy Integral Formula . Let’s see why it’s true. Let  0 be any positive number. We know that f is continuous at z 0 andsothereisa number such that | f z f z 0 |  whenever | z z 0 | . Now let 0 be a number such that and the circle C 0 z : | z z 0 | is also inside C . Now, the function f z z z 0 is analytic in the region between C and C 0 ; thus C f z z z 0 dz C 0 f z z z 0 dz . We know that C 0 1 z z 0 dz 2 i ,sowecanwrite C 0 f z z z 0 dz 2 if z 0 C 0 f z z z 0 dz f z 0 C 0 1 z z 0 dz C 0 f z f z 0 z z 0 dz . For z C 0 we have f z f z 0 z z 0 | f z f z 0 | | z z 0 | . Thus, 6.1
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C 0 f z z z 0 dz 2 if z 0 C 0 f z f z 0 z z 0 dz 2  2 . But is any positive number, and so C 0 f z z z 0 dz 2 if z 0 0, or, f z 0 1 2 i C 0 f z z z 0 dz 1 2 i C f z z z 0 dz , which is exactly what we set out to show. Meditate on this result. It says that if f is analytic on and inside a simple closed curve and we know the values f z for every z on the simple closed curve, then we know the value for the function at every point inside the curve—quite remarkable indeed. Example Let C be the circle | z | 4 traversed once in the counterclockwise direction. Let’s evaluate the integral C cos z z 2 6 z 5 dz . We simply write the integrand as cos z z 2 6 z 5 cos z z 5  z 1 f z z 1 , where f z cos z z 5 . Observe that f is analytic on and inside C ,and so, 6.2
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C cos z z 2 6 z 5 dz C f z z 1 dz 2 if 1 2 i cos1 1 5 i 2 Exercises 1. Suppose f and g are analytic on and inside the simple closed curve C , and suppose moreover that f z g z for all z on C . Prove that f z g z for all z inside C . 2. Let C be the ellipse 9 x 2 4 y 2 36 traversed once in the counterclockwise direction. Define the function g by g z C s 2 s 1 s z ds . Find a) g i b) g 4 i 3. Find C e 2 z z 2 4 dz , where C is the closed curve in the picture: 4. Find e 2 z z 2 4 dz ,whe re is the contour in the picture: 6.3
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6.2. Functions defined by integrals. Suppose C is a curve (not necessarily a simple closed curve, just a curve) and suppose the function g is continuous on C (not necessarily analytic, just continuous). Let the function
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This note was uploaded on 10/27/2009 for the course MBA 1918272 taught by Professor Peter during the Fall '09 term at Aberystwyth University.

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ch6 - Chapter Six More Integration 6.1. Cauchys Integral...

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