(a) The Taylor series expansion of e^3z about z=0 is 1+3z+ 9/2 z^2 + 27/6z^3 +….
(b) The Laurent series expansion of f(z)= e^3z/z^3 about z=0 is 1/z^3 + 3/z^2 + 9/2z+ 27/6+….
(c) C−4 = 1/72,C−3= 1/6,C−2=− 1/2,C−1=3,C0=-20, C1=9.
(a) The Taylor series expansion of e^3z about z=0 can be found by taking derivatives of e^3z at z=0. The expansion starts with the constant term 1 and includes powers of z multiplied by the corresponding derivatives of e^3z.
(b) The Laurent series expansion of f(z)= e^3z/z^3 about z=0 involves expressing the function as a sum of terms with positive and negative powers of z. In this case, it begins with the term 1/z^3 and includes higher-order terms with coefficients determined by the derivatives of e^3z.
(c) From the Laurent series expansion in part (b), the values of the Laurent coefficients are as follows:
C−4 = 1/72,C−3= 1/6,C−2=− 1/2,C−1=3,C0=-20, C1=9.
These coefficients represent the coefficients of the terms with corresponding powers of z in the Laurent series expansion.
Complete ques:
Consider the complex function f(z) = e^3z/ z^3
(a) Find the Taylor series expansion of e 32 about the point z = 0.
(b) Find the Laurent series expansion of f(z) about the point z = 0.
(c) From the expansion that you found in part (b), state the values of the Laurent coefficients C-4, С-3, С-2, C-1, Co, and c₁.