Complex Analysis: Theory and Applications · Problem 5.2.5

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · logarithm-branches

Problem

Refer to Equation (5.20) and pick an appropriate value for α\alpha so that the branch of the logarithm log⁡α(z)\log_\alpha(z) will not be analytic along the ray that begins at the origin and passes through z0z_0, where: ((a)) z0=1z_0 = 1; ((b)) z0=−1+i3z_0 = -1 + i\sqrt{3}; ((c)) z0=iz_0 = i; ((d)) z0=−iz_0 = -i; ((e)) z0=−1−iz_0 = -1 - i; ((f)) z0=3−iz_0 = \sqrt{3} - i.

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