Question

4. The function in the extended complex plane is given by s(e) a) Find and characterize all the singular points of the function b) Find all Laurent series of f(z) with center zo = 0 c) Evaluate the integral f(z)dz ford: +-, counterclockwise d) Evaluate the integral s(-)d for C: -52, counterclockwise e) Evaluate the integral( f(z)dz for C:너_1, clockwise. 4

Answer 1

The given function is f(z) = e^z + z/z^3 - z f(z) = e^z + z/z^3 - z The singular points of the function f(z) are given by z^3 - z = 0 Rightarrow z(z^2 - 1) = 0 Rightarrow z(z - 1) +(z + 1) = 0 Rightarrow z - = , 1, - 1 now lim_z Rightarrow z f(z) = lim_z Rightarrow z (e^z + z)/(z - 1)(z + 1) = e^0 + 0/(0 - 1)(0 + 1) = 1/- 1 Therefore lim_z f(z) = - 1 notequalto 0 Hence z = 0 is a pole of order one of the function f(z) Again lim_z Rightarrow 1 (z - 1) f(z) = lim z Rightarrow 1(e^z + z)/z (z - 1)(z + 1) lim_z Rightarrow 1 (e^z + z)/ z(z + 1) = (e^1 + 1) /