Expand the function f(z)=log 1+Z/ 1-Z in taylor series
Expand the function f(z) = (z−1)/(3−z) in a Taylor series centered at the point z_0 = 1. Give the radius of convergence r of the series.
Let z and w be non-zero complex numbers such that zw /=1. Prove that if z= z^(-1) and w=w^(-1),then (z + w)/(1+ zw) is real.I know z * z^(-1) = 1.
5. For z, w E C, show the following identities. (a) z + w = z + W (b) zw = zw (c) |zw| = |2||w (d) () = 1 where w #0 (e) [2"| = |z|" where n is a positive or negative integer
(5). This problem involves the mapping w(z)-,(z + z") between the z-plane and the w-plane. The two parts can be solved independently. 2 (a). Identify all of the values of z for which the mapping w(z) fails to be conformal. In each case, explain why the mapping is not conformal at that value of z. (b). Find the image in the w-plane of the unit circle Iz1, Graph it, label the axes, and label the w-plane points that correspond to...
1+ z Expand the function f(z) = in a Taylor Series Centered at Zo=i. Write the full series i.e., all the terms. Use The Sigma Notation. Find the radius R of the Disk of Convergence centered at zo.
Prove that Z [i] satisfies the definition of Euclidean Domain : W/Z = N(W) LN(z)
for the sample space {w,x,y,z}, p(x)=0.2, p(y)=0.15, p({w,y})=0.7, p({x,z})=0.3. Find p(w), p(z), and p({w,x,z}), using the properties of probability.
For each pair of vectors z and w in C2, we define an inner product(z, w〉 on C2 as follows Prove that for any z-(a, 22)" є c2, we have 2112 +12S (z, z) S 31(112)
23. Consider the function w(z) = 2-2 (a) Where in the complex z-plane are the poles of w(z)? (b) Determine the first three terms for the Taylor series expansion of w(z) about 0 (c) Identify the region of convergence for the Taylor series of part (b). (d) Determine the general expression for the n'h coefficient of the Taylor series expansion of part (b) 208 INTRODUCTION TO COMPLEX VARIABLES (e) There is a Laurent series expansion for wC) about-= 0 in...