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# Calculus and AnalyticGeometry MATH 231

UWM

GPA 3.64

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This 5 page Class Notes was uploaded by Destiny Heaney on Tuesday October 27, 2015. The Class Notes belongs to MATH 231 at University of Wisconsin - Milwaukee taught by Staff in Fall. Since its upload, it has received 52 views. For similar materials see /class/230274/math-231-university-of-wisconsin-milwaukee in Mathematics (M) at University of Wisconsin - Milwaukee.

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Date Created: 10/27/15

The complex exponential function 1 Comment You will not need this material in Math 231 but you will need it in later course in mathematics physics and electrical engineering 2 Why a complex exponential function should exist Recall that by de nition expr lim 1 1 Hoe n where r is any real number It will be shown in Math 232 that i T2 T3 r exprnliar 101r mt We could do this now with the binomial theorem but this topic is not a part of this course In either case it is the case that we could consider 7 to be a complex number and have an in nite sequence of complex numbers In the case where r it where t is a real number and 2392 71 we would get t2 t4 th expz t 1 71 gt 7 E 4 2n t3 t5 thil 39 1 t 1 Zn5 olt 3W5 H 27171 ft 290 The existence of the two limits which I have called ft and gt is easily established Via the theorem on limits of monotone functions and we can show that W 1 90 0 f t 7W g t ft all of which allows us to conclude that ft cost and gt sint as pretty remarkable result Theorem 1 For each real number t i it n i expzt 11m 1 gt cost zs1nt Hoe n For example exp2m39 l It follows from the addition formulae for sine and cosine that for any two real numbers It and s that exp2t 25 coss t 2 sins t cost coss 7 sint sins 2 sint coss sins cost cost 2sint gtlt coss 2sins exp2t exp2s so the exponential function property is still valid In fact7 for any complex number a b27 we may de ne expa b2 expa expb2 and we get an exponential function de ned on all complex numbers By this we mean that exp expy expx y for z and y complex numbers7 not just real numbers 3 Applications to trigonometric identities We have for any real numbers A and B cosA cosB 7 sinA sinB 2 sinA cosB sinB cosA 7 cosA 2sinA cosB 2sinB exp2A exp2B expA B2 cosA cosB 7 sinA sinB 2 sinA cosB sinB cosA cosA B 2sinA B so the identity exp239A exp2B exp2A B encapsulates both the sine and cosine addition formulae ln fact7 it follows by induction that cosA 2sinAN cosNA 239sinNA for any real number A and any integer even negative integers N For example7 if we want to nd the triple angle formulae for sine and cosine cos3A 2sin3A cosA 2sinA3 cos3A 32 sinA cos2A 7 3 sin2A cosA 7 2sin3A cos3A cos3A 7 3sin2A cosA sin3A 3sinA cos2A 7 sin3A Observe that we also have COSltAgt exp2A exp72A 2 exp2A 7 exp72A sinA 22 We can then see that cosA cosB iexpz A exp7z39AeXpz39B exp7z39B expz A B eXp7z39A B expz B 7 A eXp7z39A 7 wgt74gtgt7 cosA B cosA 7 B sinA sinB 7iexpz A 7 exp7z39AeXpz39B 7 exp7z39B 7iexpz A B eXp7z39A B 7 expz39B 7 A eXp7z39A 7 cosA 7 B 7 cosA B In particular sin sinkz cos ltkx 7 7 COS 1l 1 Similarly 3mm 0033 7 5mm 7 explt7 Agtgtltexplt Bgt epo39B ltexplt ltA B 7 explt7 ltA B expw 7 B 7 SXPH V BM 5mm B mm 7 B sinA B 7 sinB 7 A sin coskz sin 1x 7 7 Sin ltkx 7 2 Another application a bit fancier is the following Suppose that 005A 7 139 Then N71 N71 1 coskz Z39Sinkz 7 COSW ZSula k 17 cosx z sinxN 1 7 cosz Z39Sinx 17 cosNx 7 Z39SinNx 1 7 cosx 7 Z39Sinz 17 cosNx 7 Z39SinNx 17 cosx 7 Z39Sin 1 7 cosx 7 Z39Sinz 1 7 cosx Z39Sin 1 7 eXpZ39Nx17 exp7z39x 1 7 cos2 sin2x 17 eXpZ39Nz 7 exp7m expz39N 71 21 7 cosx N71 17 cosNx 7 cosx cosN 71 lg COSU 7 21 7 cosx 1 7 cosx 7 cosN 71 2 4sin2z2 N l 7 sinNx sinz sinN 71 k0 smUm 7 217 cosx i 7 sinNx sinz sinN 7 1x 4sin2x2 Note that these identities could also be derived from 1 and 4 The relation to the geometric properties of complex numbers Recall that we may interpret a in as a point in the place corresponding to the point 11 From the Pythagorean Theorem and the de nition of absolute value as the distance from a number to 0 we see that la btlz a2 b2 a b a 7 in so la Va 192 la 7 Ml Recall that the number a 7 in is called the complex conjugate of a 7 in If la 1921 31 0 then ab lab llt gt lab l cos6 Z39sin6 a b la bz39l la bz39l where 6 is an angle measured in radians please from the ray joining 0 to 1 to the ray joining 0 and 1 in We usually choose 0 S 6 lt 27139 but we don7t have to Once you choose 6 you can replace it by 6 2717f where n is any integer Now that we have the complex exponential function we see that we can write any non zero complex number a in as expc 26 where 6 is as above and c lnla Ml For example 1 1 7T 1 ex ltln gt p 4 5 Derivatives If C is any complex number it is easy to check that d E expcx cexpcz by proceeding in two steps First write 0 a in so expcz expaz expz bx If we can di erentiate the second term then we can apply the product rule d d E expz bx cosb z sinbz 7b sinb 2b cosb z bz sinb cosbx z b expb 4 Therefore expcz expa expz bz a expaz expz bx expax z b exp 21m a in expaz expz bx c expcx For example7 if fx exp2 3 z then f 2 3239 exp2 30

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