4. For the function2-12³ (z − 2)1(4)find the first few terms of the Laurent series about the origin, that is, oneseries for each annular ring between singular points. Find the residue ofthe function at the origin. (Warning: To find the residue, you must use theLaurent series which converges near the origin.) Hints: See Boas' Prob-lem 14.4.2. Use partial fractions as in Boas' Eqs. (14.4.5) and (14.4.7).Expand a term 1/(z − a) in powers of z to get a series convergent for|z| < a, and in powers of 1/z to get a series convergent for |z| > a. Dothis analytically. Ok to check with Mathematica.

Since we know that residue at origin means at z=0 is coefficient of 1/z.

Answered: 4. For the function 2-1 z³(z − 2) 1 (4)…

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