Question

Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7]

Answer 1

solution) in then it must. u x- y² + 2n you of the function is hormonic satisfica Leplace equation ou = -241 so the 2 -2 =0 function is harmonic ve elasy the tunition is harmonic then -csiny Starr COM NR. JAN Celebrate Lite .... Celebrate Festivals

e soy- és csy = 0 so tra fortion is harmonic - Who have to prove that f(2)= 2² + 2 is andytic put z = xtiy f(2)= (xtiyļ? txtiy f(2)= x² + ² y ² thring &xtiy -_ = -y? + + zwiy tiy = ulary) & iufwy) ulary)= x² - y² + x lay)= 2xy + y for the function to be andytic - CR equation should be satisfied au zat V

Here au oy and g ove so the function is analytic f(2)= one thing f'(2)2 (2_n +1 ) + i zy - antziy tl 24 atiy) + ) = 22 + - Ane : 3 u= nyt atzy-5. I the function is hermonic ou + d2y = 0 then 42 au On o duzo OM? Sunday 09 d =0 2, Oy prumu dry 2017 so the finition is harmonic A SRL 88 ON exow Celebrate Illa

Here we have to find it's cougate function We know that - f(z)e utiv - v= vcnly) (CR equations) au - OV Du du = -au ide + au day On integreting both sides r. Jag du + for dy v = -((x+2 ) dx + f(y + 1) dy v = (x2 + 2x) + y² ty I v- x 2 - 2x + y + y pril 201