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This 1 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 214 at University of Kentucky taught by Peter Hislop in Fall. Since its upload, it has received 8 views. For similar materials see /class/228144/ma-214-university-of-kentucky in Mathematics (M) at University of Kentucky.
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Date Created: 10/23/15
MA2147003 Fall 2008 Some Useful Results Complex Numbers7 Exponentials7 and Euler s Formula September 267 2008 Exponential and Log Functions 1 eaeb eab eab 6a eba 5 log 1091 12 gt 0 logab loga 7109b ab gt 0 logab loga 1091 ab gt 0 P FP DN ab away a gt 0 Complex Numbers 1 De nition 2 ziy z y real numbers We call I Rez7 the real part of z and y lmz7 the imaginary part of 2 Note that both I and y are real The imaginary number i is i so that i2 71 Sometimes engineers write j for i E0 Addition Let 21 I1 iy1 and 22 2 W27 then 21 22 I1 12 iyi 92 9 Multiplication As above7 2122 1112 7 mm iIiy2 12 F Complex Conjugation If 2 z iy then its complex conjugate E is E z 7 iy Sometimes the complex conjugate is denoted by 2 5 Modulus For 2 z iy we have M2 52 12 y2 This is a real number 533 Euler s Formula For a real number 6 we have em 0089 isim This can be proved using the expansion 00 J u e 7 j 7 j0 and the Taylor expansions for the sine and cosine functions 7 A Related Formula For 2 z iy we have ez ex y exely 616083
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