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# Class Note for MATH 322 with Professor Glickenstein at UA

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This 3 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 14 views.

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Date Created: 02/06/15
Chapter 13 Complex Numbers Sections 135 136 amp 137 Chapter 13 CmnplE Numbers Dehmuon tiea o The exponential of a complex number 2 x iy is defined as expz eXPXy eXPXeXPiy expX COW 139 siny 0 As for real numbers the exp derivative ie dz 0 The exponential is therefore onential function is equal to its 1 expz expz 1 entire 9 You may also use the notation expz ez Chapter 13 CmnplE Numbers o The exponential function is periodic with period 27ri indeed for any integer k E Z expz 2k7ri expx cosy 2k7r isiny 2k7r expx cosy isiny expz 0 Moreover leXPZl leXPXl leXP l eXPX COSZM sin2y expx exp lRez a As with real numbers a exp21 22 exp21 exp22 o expz 75 0 Chapter 13 CmnplE Numbers Trigonomelrlc and b C 2 Trigonometric functions 0 The complex sine and cosine functions are defined in a way similar to their real counterparts eiz eiiz cosz 2 IZ eilz sinz 2 o The tangent cotangent secant and cosecant are defined as usual For instance sinz tanlz cosz7 secz L etc cosz Chapter 13 CmnplE Numbers 1 quotemal Trigonomelrlc and h Tn gonom emc functions vaerholln Inntiuns Trigonometric functions continued 0 The rules of differentiation that you are familiar with still work 0 Example 9 Use the definitions of cosz and sinz eiz eiiz Biz 2 7 to find cosz and sinz cosz a Show that Euler s formula also works if 0 is complex c fl I nc non IS functions I lunctu en Hyerbolic functions o The complex hyperbolic sine and similar to their real counterparts cosine are defined in a way eZ e Z Z Z 2 7 sinhz e e coshz 3 o The hyperbolic sine and cosine as well as the sine and cosine are entire 0 We have the following relations coshiz cosz7 sinhiz i sinz7 4 cosiz coshz7 siniz i sinhz on 939 Nu m b ers Chapter CmnpleyNunibers Chapter a Dennmon pnm Ipal value 2 M iii 4 Complex logarithm o The logarithm w of 2 0 is defined as eW z 0 Since the exponential is 27ri periodic the complex logarithm is multi valued 0 Solving the above equation for w w M and z rem gives W 7 eW eW elW39 rem gt e r T r 7 w 0 2p7r which implies w lnr and w 0 2p7r p E Z 0 Therefore lnz lnlzl i argz 7 ple Numbers Chapter reinn Ion prlnclpal value of lnl lzll Principal value of lnz 0 We define the principal value of lnz Lnz as the value of lnz obtained with the principal value of argz ie Lnz lnlzl I39Argz 0 Note that Lnz jumps by 727d when All312 one crosses the 0 X negative real axis Argz gtTc from above 0 The negative real axis is called a branch cut of Lnz 939 Numbers Definition Principal value of im lzll 3 Recall that Lnz lnlzl I39Argz 0 Since Argz argz 2p7r p E Z we therefore see that lnz is related to Lnz by lnz Lnz i2p7r p E Z 0 Examples 0 Ln2 ln2 but ln2 ln2 i2p7r7 c Find Ln74 and ln74 0 Find ln10i pEZ Chapter 13 Compier Numbers Deiuu Ion prlnclpal value of lnl lzll Tm lunctu en Properties of the logarithm a You have to be careful when you use identities like 21 ln2122 ln21ln22 or In lnzliln22 They are only true up to multiples of 27ri o For instance if 21 i expi7r2 and 22 71 expi7r ln21 i 2p1i7r ln22 i7r2p2i7r p1p2 E Z and 3 ln2122 i rls2p3i7r7 pg 62 but p3 is not necessarily equal to p1 p2 Chapter 13 umpier Numbers Definition Principal value of im lzll continued 0 Moreover with 21 i expi7r2 and 22 71 expi7r Ln21 ig Ln22 Mr and Ln21 22 7 y Ln21 Ln22 0 However every branch of the logarithm ie each expression of lnz with a given value of p E Z is analytic except at the branch point z 0 and on the branch cut of lnz In the domain of analyticity of lnz d 1 Ellnlzll 2 Chapter 13 Compier Numbers 5 Definition ump r um n 5 Complex power function o lf 2 0 and c are complex numbers we define zc expclnz exp anz2pc7r p62 0 For c E C this is again a multi valued function and we define the principal value of 25 as zc exp c Lnz Chapter 13 umpier Numbers

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