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# Class Note for MATH 322 with Professor Glickenstein at UA 2

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Date Created: 02/06/15

ruww 0 Complex numbers are of the form zxiy xyER wx sumer WETquot 37L o In the above definition X is the real part of z and y is the imaginary part of z o The complex number 2 X iy may be represented in the complex plane as the point with cartesian coordinates xy evumrmnw o The complex conjugate of z x iy is defined as 2 x 7 I39y 0 As a consequence of the above definition we have 92 22 Smz 22 x2 y2 1 o If 21 and 22 are two complex numbers then 2122Z1Z27 2122372 2 Imam WWW o The absolute value or modulus of z x iy is lzl V22 X2 y2 It is a positive number 0 Examples Evaluate the following 0 W a i273i new We DJ mu Iml 39 mun a You should use the same rules of algebra as for real numbers but remember that i2 71 Examples 0 1311 Find powers of i and 1i 0 Assume 21 2 3 and 22 71 7 7 Calculate 2122 and 21222 0 Get used to writing a complex number in the form 2 real part i imaginary part7 no matter how complicated this expression might be 1 39 hn quotIons 2 Remember that multiplying a complex number by its complex conjugate gives a real number 0 Examples Assume 21 2 3 and 22 717 7i 21 c Find 22 0 Find 2 22 1 c Find 3m o 13227 Solve z2 7 8 7 5iz 40 7 20 0 l marwmmmrmm 0 ln polar coordinates X rcos07 y rsin07 where r X2 y2 o The angle 0 is called the argument of 2 It is defined for all 2 0 and is given by arctan if X 2 0 argz0 arctan f 7r ifXlt0andy20 i2n7r arctan 77r ifXlt0andylt0 Frlna jm 0 Because argz is multi valued it is convenient to agree on a particular choice of argz in order to have a single valued function 0 The principal value of argz Argz is such that tan Argz 7 with 7 7r lt Argz g 7r 0 Note that Argz y Argzn Jumps by 727r when 7 one crosses the x 0 7 X negative real axis Argz gtTc from above mama new ll I l 0 Examples 9 Find the principal value of the argument of z 1 7 i 0 Find the principal value of the argument of z 710 y 7 0 7 X 1 39hnlflons lumbar mmbzrs 0 You need to be able to go back and forth between the polar and cartesian representations of a complex number 2 X iy izi cos0 iizi sin0 o In particular you need to know the values of the sine and cosine of multiples of 7r6 and 7r4 7r 7r 0 Convert cos I sIn to cartesian coordinates a What is the cartesian form of the complex number such that lzl 3 and Argz 7r4 mm mummies o Euler s formula reads expi0 cos0 isin0 0 E R 0 As a consequence every complex number 2 0 can be written as z izi cos0 isin0 izi expi0 o This formula is extremely useful for calculating powers and roots of complex numbers or for multiplying and dividing complex numbers in polar form was To find the n th power of a complex number 2 0 proceed as follows 9 Write z in exponential form 2 izi exp 0 G Then take the n th power of each side of the above equation 2 izi exp in izi cosn0 isinn0 Q In particular if z is on the unit circle 1 we have cos0 isin0 cosn0 isinn0 This is De Moivre s formula 391 slinilmiis Al Polar coordinates l g f Iv 0 0 Examples of application 0 Trigonometric formulas cos20 c0520 7 sin207 sin20 2sin0 cos0 0 Find cos30 and sin30 in terms of cos0 and sin0 o The product of 21 r1 exp 01 and 22 r2 expi02 is 21 22 r1 exp 101 r2 exp 102 r1r2eXP 01 92 4 0 AS a consequence arg21 22 arg21 arg227 llegll21llZQl 0 We can use Equation 4 to show that cos 01 02 cos 01 cos 02 7 sin 01 sin 027 5 sin 01 02 sin 01 cos 02 cos 01 sin 02 WW WWWM Z 0 Similarly the ratio 1 is given by Z2 21 7 r1 exp 01 7 r1 22 7 r2 eXPWZ 7 E eXPl 91 7 92 0 AS a consequence 21 21 W arg arg21 7arg227 7 22 22 0 Example Assume 21 2 3 and 22 717 7 Find 22 was D iitii To find the n th roots of a complex number 2 0 proceed as follows 9 Write z in exponential form 2 rexp i0 2p7r7 with r lzl and p E Z G Then take the n th root or the 1n th power 0 2 0 2 y zun runexp Wexp M n n Q There are thus n roots of 2 given by zkquotFcoswgt isin k07 7n71 Polar coordinat Roots of a complex number continued er IIIIlIIlJf I E Polar coord In Roots of a complex number continued o The principal value of 2 is the n th root of 2 obtained by taking 0 Argz and k 0 o The n th roots of unity are given by Wcos gtisin gtwk k0n71 where w cos27rn isin27rn a In particular if W1 is any n th root of 2 0 then the n th roots of z are given by 2 n71 W17W1w7W1w77W1w quotpl3y Numbers 5 Examples 0 Find the three cubic roots of l 0 Find the four values of 0 Give a representation in the complex plane of the principal value of the eighth root of z 73 4139 Num bars Triangle inequality o If 21 and 22 are two complex numbers then lZ1 22l S lZ1l l22l This is called the triangle inequality Geometrically it says that the length of any side of a triangle cannot be larger than the sum of the lengths of the other two sides a More generally if 21 22 z1 are n complex numbers then lZ122znl l21ll22ll2nl quotpl3y Numbers

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