(a) Suppose the principal branch of the logarithm f (z) Ln z loge

Chapter 19, Problem 31

(choose chapter or problem)

(a) Suppose the principal branch of the logarithm \(f(z)=\operatorname{Ln} z=) \(\log _{e}|z|+i \operatorname{Arg} z\) z is expanded in a Taylor series with center \(z_{0}=-1+i\). Explain why \(R=1\) is the radius of the largest circle centered at \(z_{0}=-1+i\) within which f is analytic.

(b) Show that within the circle \(|z-(-1+i)|=1\) the Taylor series for f is

\(\operatorname{Ln} z=\frac{1}{2} \log _{e} 2+\frac{3 \pi}{4} i-\sum_{k=1}^{\infty} \frac{1}{k}\left(\frac{1+i}{2}\right)^{k}(z+1-i)^{k}\).

(c) Show that the radius of convergence for the power series in part (b) is \(R=\sqrt{2}\). Explain why this does not contradict the result in part (a).