Diff Equations & Matrix Alg I
Diff Equations & Matrix Alg I MA 221
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This 6 page Class Notes was uploaded by Eldon Trantow on Monday October 19, 2015. The Class Notes belongs to MA 221 at Rose-Hulman Institute of Technology taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/225073/ma-221-rose-hulman-institute-of-technology in Mathematics (M) at Rose-Hulman Institute of Technology.
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Date Created: 10/19/15
1 Complex Numbers Introduct1on Even though most engineering students have seen complex numbers and probably have had some experience with the arithmetic and algebra of complex numbers by the sophomore year it is usually the case that the exposure has been random We would like to present the structure and rules of complex numbers in an organized way and also give you a chance to do a few routine exercises so that complex numbers do not appear as mysterious as they once did In the same way that we think of real numbers as being points on the real line we think of complex numbers as being points in the plane The horizontal axis will be the usual real numbers and the vertical axis will be the purely imaginary numbers For instance numbers like 3 5 76 and 17 will be on the real axis and 5239 7239 713239 and 746239 will be on the imaginary axis Any number in the plane can then be written as a number from the real axis plus a number from the imaginary axis For example the number two units to the right of the origin and 3 units up would be 2 3239 An arbitrary complex number would be written as 2 z iy it seems that if actual numbers are used the i is written on the right but if symbols are being used the i is written on the left This form of the complex number 2 z iy is called the rectangular form and the number I is called the real part of z and y is called the imaginary part of 2 Standard notation for the real and imaginary parts of a complex number are I Rez and y lmz The symbol 239 is usually used in mathematics but in some engineering disciplines j is used since i has other meanings current for example Thinking of complex numbers as vectors can often be useful Addition and subtraction of complex numbers is the same as it is for vectors When adding or subtracting complex numbers we collect terms with respect to i 23i 57i 25 37i710i 23i7 57i 275377i7374i It s the multiplication and division of complex numbers that makes complex numbers different from vectors Since 239 71 it follows that i2 71 so when multiplying complex numbers we simplify using 2392 71 and then collect terms with respect to i For example 23i57i1014i15i21i2 1014i15i721 71129i Division on the other hand can be a bit more challenging Here we make use of the fact that the product of a complex number with its conjugate E 1 fig is 12 32 a real number In other words 2 ziyziiyz2fizyiyziiyiy12y2 When dividing a complex number 2 by another complex number w we multiply the numerator and denominator by the conjugate ofw producing a real number in the 39 andacomnle 39 quot 39 in the numerator For example to divide z 23i by w 52i 23i 7 23i 572239723i572i71074i15i76i2 52i 52z 39 572i 52i572i 254 1074i15i671611i 716 11 7 29 7 E2 The magnitude length of a complex number 2 z iy is 39r dz 32 lf 9 is the angle between 2 and the positive z axis then the real part of z is Re z TCOS9 and the imaginary part is lm y Tsint9 Using 39r and t9 the complex number 2 can be expressed in the form 2 rem known as the polar form of the complex number We now have a number of ways of representing complex numbers 2 z iy 7 cos6 ir sin6 7 cos9 239 sin6 29 7 6 where we have use Euler s Formula em cos6 239 sin6 We will justify Euler s Formula later Notice that multiplication of two complex numbers in polar form is easy lfz 563i and w 76 then zw 56 i7 4i 5 7 63139 64139 3563i4i 356 the T s are multiplied and the 9 s are added Exercise 1 If 2 2 3239 and w 4 7 22 calculate the following a 22 b 2w c d 32 7 5w e Rez f lmw g E h lwl i 22 111E k 2w l zw The conjugation operation can be used in a variety of ways For instance ifz and E are added the result is 2Rez that is 2 I I 7 21 In the same manner you can show that the difference of z and E is 2239 lmz Furthermore it s easy to see that the conjugate of a sum is what it should be ie the sum of the conjugates and it s remarkable that the conjugate of a product is also what it should be ie the product of the conjugates The next exercise asks you to verify these results by doing some examples ExerciseZ fz24z andw52i a showz EE b showzw m In addition to adding subtracting multiplying and dividing complex nume bers we can also evaluate functions of complex numbers As long as the function is algebraic we can use our basic rules Exercise 3 Let fz 22 Calculate f132 Convert both 132 and f132 to polar form How are the magnitudes related How are the 639s related As another example let 22 32 Evaluating f requires multiplying and adding For example f2 3239 2 3i2 32 3239 l 21239 Even though this function is more complicated it is still algebraic and the evaluation can be done fairly easily The problem comes when the function is not a simple algebraic function For instance what is lnz sinz or even Some of these functions are difficult to analyze and in fact a detailed study of these types of functions is usually done in an upper level mathematics course devoted to complex variables Since the nonealgebraic function that is most often used in engineering and science is the exponential function 62 we will concentrate on its properties and you will see these properties being applied to our study of second order differential equations The algebraic properties for ez are the same as for 6 That is ezw 62610 60 l 1 5 2 e Euler s formula 6 0089 isin allows us to write 62 in rectangular form For example e2 ew ly ewe e cosy 239 sin e cosy ie sin The term 6 is the magnitude and 3 determines the angle Notice that since cosy and sin are periodic 62 will also be periodic in the variable 3 For instance es227ri 32i27ri 32i 62m 32i Also note that adding 7139 to the imaginary part will simply change the direce tion of 62 exiltym eaciyi7r awnem eaciylt71 iexiy The next two exercises ask you to evaluate and plot a variety of complex numbers of the form 62 Note that the number 619 will always be on the unit circle and that the conjugate of 619 is 649 That is eW cos 6 isin 6 cos 6 7 isin 6 cos7 9 isin 76 e49 Note that we used two facts from trigonometry 00879 0089 and sin 79 7 sin A similar argument shows that the conjugate of e is 67 Exercise 4 Evaluate and plot on one set of axes the following oomplex num bers eOz 27r1 671 em 63m ezl 2n7rz ET e An arbitrary complex number of the form 62 61 Will not be on the unit circle unless I 0 For example7 1 522 52239 2 7r 1 El EQETZ e2 cos 2 sin 2 e2 i lt4 lt4 So When determining the placement of 61 in the complex plane7 use I to determine how far the complex number is from the origin7 and use 3 to determine the angle the complex number makes With the positive real axis The real and imaginary parts of 61 are also easy to determine Ree iy ea cosy and lme iy e sin Exercise 5 Evaluate and plot on one set of axes the following oomplex num bers 3 127r1 1Tz 671 0537rz 17 e2 672 E Ti 6700 Exercise 6 a Find a z for which e2 6239 b Find a z for which e2 4239 c Find 2 2395 for which e2 4 d Find a z for which e2 1 2239 Now that we ve had some practice working with Euler s formula let s look at why it s true One way to prove this relationship between the exponential sine and cosine functions is to make use of Taylor series which were introduced in calculus If we assume and we will that the Taylor series for 62 has the same form for complex numbers as for real numbers then 2 3 4 5 6 19 1i9w 92 93 94 95 96 97 98 1zt975752EEziai z 92 94 96 93 95 97 1 EZ Eml9 E cos6 239 sin6 The proof isn t long or difficult it just requires some knowledge of the Taylor series expansion for the exponential sine and cosine functions 1s section conta1ns an example related to solutions of linear second order differential equations As seen in class when we arrive at the solution to a linear second order differential equation which has complex numbers the graph of the solution appears to be a sine or cosine curve which is possibly decaying Also when using dsolve in Maple complex numbers do not appear Application to Differential Equations T Where did all the complex numbers go We give an example showing how to convert the complex form of solutions to a form which has only real numbers Even though the following example is a bit messy and technical it does illustrate how Euler s formula and some complex arithmetic are used to eliminate the complex numbers which initially appear in solutions Example 2 When solving 01 15 4175 0 10 0 z 0 l the characteristic equation is 39r 4 0 which has solutions 39r iZi Therefore the general solution is 175 0162 62672 The initial conditions give the equations cl Cg 0 223901 7 223902 1 whose solution is cl 71202 Please notice in this example and in the others that cl and Cg are conjugates Therefore the complex form of the solution is l l 175 71262 Zie t If we were to plot 175 in Maple we would see a sin curve In other words even though there are complex numbers in the expression the complex numbers seem to disappear when plotting Euler s formula comes to the rescue Mt iiiem iieim iii cos2t 239 sin2t cos72t isin72t iii cos2t 239 sin2t cos 2t 7 isin2t l l l l 712 cos2t Z s1n2t 12 cos 2t Z s1n2t sin2t
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