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Class Note for MATH 254 at UA

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 23 views.

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Date Created: 02/06/15
MATH 254 Complex Analysis Complex numbers arose because polynomial equations With real coefficients did not always have real rootsl elgl az2 121 c 0 1 7 i l lf 22 lt 4ac7 then there are no real roots because 2a 2a t l real real a new kind of object We call 5 i so that i2 71 De nition A complex number 2 z iy is an ordered pair of real numbers 17y and is Written as I iyl We picture it as a point in the plane y pure imaginary r numbers y 6 X V X pure real numbers lts length7 Written as is xz2 y We call 3 z 7 iy the complex conjugate of 2 Addition 21 22 I1 iy1 2 iy2 11 12 iy1 y2 Mllltiplication 2122 11 iyi12 23992 1112 y1y2 i11y1 I2y2 m 2122 11 iy1z2 ZQ2 1112 y1y2 iIiyi z2y2 2 Ex Write 6 Z in the form a ii a 12 real It is convenient to Write 2 z iy Tcost i7quot sint in polar representation 7 xz2 y2 and 9 is such that 3086 sint9 We call 9 by argzl It is only determined up to 27L eg 2 l 7 7 cost9l7 Tsln67 gtT27 3086 sin97 97 2k7r7 160 i1 ill lfwe stipulate 7 lt 9 S 7r7 then for a given 2 the 9 is unique and is called Arg 2 lf 9 is in radians7 by Taylor series 92 94 93 65 COSOZSID917 Einz67557 3 4 5 1z 6 26 miew 2x 3 4x 5 So 3086 isint 6119 Then cost9 7 isint e49 Add and divide by 2 gt 0086 eigigflg 7 Sing 819722 Subtract and divide by 2i Also cos 9 i sin 9 619 Bing cos m9 isin m9 Set n 2 308267 sin2t9 308267 2sint9cost9 sin2t9 Set n 3 and express cos3 9 cos 39 icos 9 sin3 9 7 Roots of complex numbers Suppose you want to nd 2 Where n is an integer Write w 2 or w 2 Let w pawl Now Write 2 T i92k7ri Where 9 Argz the principal argument iiei 7 lt 9 S W ThenpT andg02k7r7 k07l727iunili n Ex 1 2liz1i2 i2k7riw3p363w p2 7 g02k7r k07l72i T 7 cos31isin31 4 4 7 7 2611 2 ltcos112isingt 7 2 M as Cube roots of l i are 2ei7 263i cos 12 2s1n12 i Ex 2 cub A 2 k u L A u m 391 391 391 704 2721i e 22kmi714 l le 8 2 7 k07l7273e 187 6318 E7137 elhsi Note Is 4 gives e2m e i So the values repeat Ex 3 l 77r 39 1 77r 39 77r I i Z 22 2 e 2 21 27 eTkm7 k 071 ieT s1nce e7 fl 753

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