Mathematics of Physics and Engineering
(^206) Power Series 4.3 Power Series and Analytic Functions 4.3.1 Series of Complex Numbers A series of complex numbers ck, k &g ...
Series of Complex Numbers 207 The following result is known as the ratio test for convergence. Theorem 4.3.1 Define „ ,. Icn+l| ...
208 Power Series Proof. If M < 1, then, by the definition of limsup, there exists age (M, 1) and a positive integer N so that ...
Convergence 209 Proof. Define z = z — ZQ. For z ^ 0, apply the root test to the resulting series (4.3.1): K = limsup \anzn\^n = ...
(^210) Power Series EXERCISE 4.3.7.c Find the radius of convergence of the power series 7.2n+^1 (dnV E(5+(_ln« fc>l Let us no ...
Convergence 211 where By (4.3.6), the radius of convergence of the power series in (4.3.8) is at least p. This shows that an ana ...
212 Power Series The value of \z — zo|/|z — C| is constant for all £ on Cp and is less than one, which allows you to make the su ...
Exponential Function 213 of z. Differentiating term-by-term the power series for g gives f{z) = Ek>i(-l)k+lk(z ~^1 )k~^1 = Ef ...
(^214) Power Series Euler's formula eie = cos6 + ism6, (4.3.14) and the relations eiz — e~*z eiz 4- e~iz sinz= — , cosz=. (4.3.1 ...
Laurent Series 215 is specified by the condition Lnl = 0. More generally, by fixing the value of ln z at one point ZQ ^ 0, we de ...
216 Singularities of Complex Functions of ZQ. The proof of this series representation essentially relies on the fact that a neig ...
Laurent Series 217 where Ck = :ti(f '» '-^fc+idC, fc = 0,±l,±2,..., (4.4.3) 27riJCo(C-z 0 )fe+1' and Cp is a circle with center ...
(^218) Singularities of Complex Functions Then _m = f /(OK-)- 1 (447) and it remains to integrate this equality. Step 5. We comb ...
Laurent Series 219 lytic at ZQ and there exists a S > 0 so that the function / is analytic in the region {z : 0 < \z — ZQ\ ...
220 Singularities of Complex Functions EXERCISE 4.4.2.C (a) Let ZQ be an isolated singular point of the function f = f(z), and c ...
Laurent Series 221 EXERCISE 4.4.3.c Find the Laurent series for the function (2z + 5\ f(z) =JK cos ' \z + 2 J around the point Z ...
222 Singularities of Complex Functions of / at zo = 2; the Laurent series at the point ZQ = 2 must converge when 0 < \z — 21 ...
Residue Integration^223 where C(zo) is a simple, closed, piece-wise smooth curve enclosing the point zo and oriented countercloc ...
(^224) Singularities of Complex Functions singularity. Computation of the residue at a removable singularity or a pole does not ...
Residue Integration 225 terms after (TV — l)!c_i disappear as well. As a result, 1 / dN~l \ In particular, if f(z) = g(z)(z — ZQ ...
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