Taylor and Laurent Series

Transcription

1 hapter Nine Taylor and Laurent Series 9.. Taylor series. Suppose f is analytic on the open disk z z 0 r. Let z be any point in this disk and choose to be the positively oriented circle of radius, where z z 0 r. Then for s we have s z s z 0 z z 0 s z 0 zz 0 sz 0 j0 z z 0 j s z 0 j since zz 0 sz 0. The convergence is uniform, so we may integrate fs s z ds j0 fs ds j s z 0 z z 0 j,or fz 2i fs s z ds j0 2i fs s z 0 j ds z z 0 j. We have thus produced a power series having the given analytic function as a limit: fz c j z z 0 j, z z 0 r, j0 where c j 2i fs ds. j s z 0 This is the celebrated Taylor Series for f at z z 0. We know we may differentiate the series to get f z jc j z z 0 j j and this one converges uniformly where the series for f does. We can thus differentiate again and 9.

2 again to obtain f n z jj j 2j nc j z z 0 jn. jn Hence, f n z 0 n!c n, or c n fn z 0 n!. But we also know that c n 2i fs ds. n s z 0 This gives us f n z 0 n! 2i fs ds, for n 0,,2,. n s z 0 This is the famous Generalized auchy Integral Formula. Recall that we previously derived this formula for n 0 and. What does all this tell us about the radius of convergence of a power series? Suppose we have fz c j z z 0 j, j0 and the radius of convergence is R. Then we know, of course, that the limit function f is analytic for z z 0 R. We that if f is analytic in z z 0 r, then the series converges for z z 0 r. Thus r R, and so f cannot be analytic at any point z for which z z 0 R. In other words, the circle of convergence is the largest circle centered at z 0 inside of which the limit f is analytic. Example Let fz expz e z. Then f0 f 0 f n 0, and the Taylor series for f at z 0 0 is 9.2

3 e z j! z j j0 and this is valid for all values of z since f is entire. (We also showed earlier that this particular series has an infinite radius of convergence.) Exercises. Show that for all z, e z e j! z j. j0 n 2. What is the radius of convergence of the Taylor series c j z j for tanhz? j0 3. Show that z j0 z i j i j for z i If fz z, what is f0 i? 5. Suppose f is analytic at z 0 and f0 f 0 f 0 0. Prove there is a function g analytic at 0 such that fz z 3 gz in a neighborhood of Find the Taylor series for fz sinz at z Show that the function f defined by fz sinz z for z 0 for z 0 is analytic at z 0, and find f

4 9.2. Laurent series. Suppose f is analytic in the region R z z 0 R 2, and let be a positively oriented simple closed curve around z 0 in this region. (Note: we include the possiblites that R can be 0, and R 2.) We shall show that for z in this region fz j0 a j z z 0 j j b j z z 0 j, where a j 2i fs ds, for j 0,,2, j s z 0 and b j 2i fs ds, for j,2,. j s z 0 The sum of the limits of these two series is frequently written fz c j z z 0 j, j where c j 2i fs, j 0,,2,. j s z 0 This recipe for fz is called a Laurent series,although it is important to keep in mind that it is really two series. Okay, now let s derive the above formula. First, let r and r 2 be so that R r z z 0 r 2 R 2 and so that the point z and the curve are included in the region r z z 0 r 2. Also, letbe a circle centered at z and such thatis included in this region. 9.4

5 is an analytic function (of s) on the region bounded by, 2, and, where is the circle z r and 2 is the circle z r 2. Thus, Then fs sz 2 fs s z ds fs s z ds fs s z ds. fs (All three circles are positively oriented, of course.) But sz ds 2ifz, and so we have fz 2 fs s z ds fs s z ds. Look at the first of the two integrals on the rihgt-hand side of this equation. For s 2, we have z z 0 s z 0, and so Hence, s z s z 0 z z 0 s z 0 s z 0 j0 j0 zz 0 ss 0 z z 0 s s 0 s z 0 j z z 0 j. j 9.5

6 2 fs s z ds j0 2 fs ds j s z z 0 z 0 j. j0 fs ds j s z z 0 z 0 j For the second of these two integrals, note that for s we have s z 0 z z 0, and so s z z z 0s z 0 z z 0 sz 0 z z 0 j0 s z 0 z z 0 s z 0 j j z z 0 j j zz 0 s z 0 j j0 z z 0 j j s z 0 j z z 0 j As before, fs s z ds j j 2 fs s z 0 j ds z z 0 j fs s z 0 j ds z z 0 j Putting this altogether, we have the Laurent series: fz 2 j0 fs s z ds fs s z ds fs ds j s z z 0 z 0 j j fs s z 0 j ds z z 0 j. Example Let f be defined by fz zz. First, observe that f is analytic in the region 0 z. Let s find the Laurent series for f valid in 9.6

7 this region. First, fz zz z z. From our vast knowledge of the Geometric series, we have fz z z j. j0 Now let s find another Laurent series for f, the one valid for the region z. First, Now since z, we have z z z. z z z z j0 z j z j, j and so fz z z z j fz z j. j2 z j Exercises 8. Find two Laurent series in powers of z for the function f defined by fz z 2 z and specify the regions in which the series converge to fz. 9. Find two Laurent series in powers of z for the function f defined by 9.7

8 fz zz 2 and specify the regions in which the series converge to fz. 0. Find the Laurent series in powers of z for fz z in the region z. 9.8