Page 1 : COMPLEX ANALYSIS, , , , , , 5.1 Basic Review of Complex Numbers, , Various representation of Complex number:, , A complex number (z) is represented as z=x+iy , where xis thereal part and y is imaginary part., , J, , The conjugate ofcomplex numberz.is represented by Z =x —iy__, , , , , , The modulus of complex number z = 2} =, , , , The complex number (z) in polar form (#,8)can also be written as, z=xtiy=re® =r(cosO+ising)., , where x=rcos@, y=rsing, , =>, , , , Properties of modulus of., Ifz, and z, are two complex numbers, then, (|e, +24] sf] +l i) |2-2:|2 |All, , 2|_ lal, iii) [2,22] =|2,] 22] (iv) =f, , Properties of argument of z:, , (D Arg (2.2) 2-00, , , , z,) = Arg.(z,)+ Arg.(z,) + Arg.(z,) +--+ Arg (z,), , i) are( = Arg (z,)- Arg (22), , Some important Relations:, (i) e®= cos@ + i sind , (ii) cosO = (e+ e“)/2, (iii) sind = (e°—e“)/2i

Page 2 : 5:29PM Fri 29 Oct, , , , COMPLEX ANALYSIS, , , , 1. Mathematical Physics (1 of 7) ®, , , , Soin:, , Soin:, , Soin:, , (e+e), , (iv) cosh0=—>— ,, , o_o, (v) sinho= a, , De-Moiver’s theorem:, (cos® + i sin8)" = cos nO + isin nO, , Complex cube roots of unity:, , x =1>x=1a=— Ene, , i, , , , , , , , such that 1+ a+? =Oand @ =1, , , , Example 1: Multiplying a complex number z by | + / rotates the radius vector of z by an angle of, , (a) 90° clockwise (0) 45° anticlockwise, , (c) 45° clockwise . (d) 90° anticlockwise, , z=re® = (I+i)z=( a") e® = V2 rel*"4), , Multiplying (1 + i) with zrotatés the radius vector anticlockwise by 45° and increases the modulus bya factor, , of J2., , Correct option is (a), , , , , , , , Example 2: 1f : =(2+3)+iV5—2 (A. is areal parameter and i= J=1), then the locus of will be, (a) circle (b)ellipse (©) parabola (@ hyperbola, z=(A+3)+iV5—A? = real part x = 2+3 and imaginary part y= y5— 2?, , = pas-2? =5-(x-3)?", , Correct option is (a), , , , Pees, , , , (Equation ofcircle), , , , Example 3: If z = x + iy, then for e* to be real., (a) y=na, n=0,+1,+2,.... (bv) x=nn, n=0,41,+2,...., , (c) yeQneyF, n=0,+1,+2,.... d) xaQnenz, n=0,t1,42,...., , e =e" see” =e"(cosy+isiny), Hence, for e* to be real, siny=0 => y=nz, n=0,41,+42,...., Correct option is (a), , Example 4: If z = x+iy, then Im (2-497 +32) is equal to, (@) 3y (b) -3y © (v-4) @ -0-4)

Page 4 : 5:30 PM Fri 29 Oct, , , , COMPLEX ANALYSIS, , Sotn:, , Soln:, , Soln:, , 1. Mathematical Physics (1 of 7), , , , , , , , , , Since, z lies in 3rd quadrant. Hence, principle value of arg (z) is ai-n --%., Correct option is (c), Example 8: If|z|=1, then for all complex numbers ‘a’ and ‘b" value of a4] is equal to, +a, (a) Va? +5? () 1 (©) = @2, @, az+b|_[e(a+2)|_|elfa+be"!, lisa ~Teval (besa, |a+oa|, +a], Now, since [ ns, , , , |a-dz|=[a+bz] | eet ace, , la + bz" a +be| |, loz +a] "few ba|, , Correct option is me, , az+b|_, bz +a|, , , , , , , , Example 9: $a is equal to,, (f,, , , , @ viel ©), , “, lei _V2e'4 1 ie, et, , Hence, =, a a, , Correct option is (b), , , , at, Example 10: For the roots of unity z=e ”, m>1. The value of 1+z+2*+---2"7! is equal to, , leztze te, , , , (sum of m’ terms of GP.), , mF, Here, z"=e " =e**! =]

Page 5 : 5:30 PM Fri 29 Oct, , , , 200 of 296, , Soln:, , Soln:, , 1. Mathematical Physics (1 of 7), , , , COMPLEX ANALYSIS, , , , Correct answer is (0), , Example 11: Imaginary part of sn"(%) is, Let z=x+iy, , : -(%) ‘ i, sin”|—]=2 = sinz=, 4, , , , Let e!? = ¢, Hence, -2=-3, t 2, , , , , , , , , Therefore, t=, , 2 tat = iz =lo (j= j, 2 5 arr, , , , Hence, imaginary part = log 2, Correct answer is (0.693) |, , , , Example 12: If|z|=1, and |: +22 + 6+8i|s, Wehave, fa 425 #25 +245 fale +b, Hence, |:?-+22+6-+ 8] s|22]4+2]z| +[6+ 84 s|af., Therefore, m= 13., , Correct answer is (13), , 5.2 Function of A complex Variable, Basic Representation:, S(2)=u(x,y)+i(%y), , Pa(xtpl=@-yy + — Qxy, day) rapt oe iigmy pt, , , , Example: / (=), , Existance of lim f(z):, , ‘The limit will exists only ifthe limiting value is independent of the path along whichz approaches =,