5

Evaluate the following integrals22_22$ dz (2+1)2(22+4)cis the circle Iz| = 5[9]4-32$ (Z(2-1)(2-2) dz _ cis the circle Iz/ = 3[6]Find the harmonic conjugate function...

Question

Evaluate the following integrals22_22$ dz (2+1)2(22+4)cis the circle Iz| = 5[9]4-32$ (Z(2-1)(2-2) dz _ cis the circle Iz/ = 3[6]Find the harmonic conjugate function of u = e-3xcos(3y). Hence show that analytic function f(z) = e-3z + ik where k is a constant [10]

Evaluate the following integrals 22_22 $ dz (2+1)2(22+4) cis the circle Iz| = 5 [9] 4-32 $ (Z(2-1)(2-2) dz _ cis the circle Iz/ = 3 [6] Find the harmonic conjugate function of u = e-3xcos(3y). Hence show that analytic function f(z) = e-3z + ik where k is a constant [10]



Answers

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises $1-24 .$
a. $\int_{0}^{2 \pi} \frac{\cos z}{\sqrt{4+3 \sin z}} d z \quad$ b. $\int_{-\pi}^{\pi} \frac{\cos z}{\sqrt{4+3 \sin z}} d z$

Okay, let's start with the partial fraction. The composition of this to do are integral. So that means we have the Z plus one over rz times Z squared plus four. Gonna equal ZB in a linear factor. We have some value for the constant numerator over the Z. And since the Z squared plus four is a quadratic, there's two terms in that numerator. There's a Z term, so be easy and there could be a constant term. So plus C on divided by the C squared plus four. And I apologize sometimes my disease and my twos what kind of like each other? So now we multiply by the common denominator to get rid of our fractions. So that just gives us the Z plus one on the left hand side on the right hand side, multiplying this first fraction, the disease will reduce and we have eight times the Z squared plus four. So a Z squared plus for a and the second fraction the Z squared plus four divides out and we've got the numerator times E. So that's B Z squared, plus Sesay. And now we can set our go terms and coefficients equal to each other. We have the square terms. Well, there's no square term on the left hand side and we've got a Pulis B Z squared on the right. We have our Z term, so Z equals this easy. So that means see, is one. And then we have our constant term, the one and the four a so one equals for a So that means a has got to be 1/4 m b has to be negative One force and we already know sees one. So now we can rewrite this. Yes. 14 in the integral, uh, 1/4 over Z plus negative 1/4 times e plus one over Z squared plus four. That's all times DX. And now we actually want to take this second fraction and split it up again into two separate fractions because right now we can't integrate it in that form. So we have and grow up 1/4 over Z mine. It's the integral of 1/4 z over Z squared plus four. Actually, all these would have been easy on if I'm separating those and then plus, you have the integral up one over Z squared pause for Do you see, uh uh, Now, to do each of these individually, first, these constants could get pulled out in front. So let me do that. Here. That 1/4. Pull that out in front and that becomes 1/4 and a girl off one over z dizzy. Hopefully you recognize by now that's just our one over you, d you. So that would be 1/4 times a natural. Uh, absolute values e For this 2nd 1 we pull out that one force again. And now, because of that Z term in the numerator, I can't let you equal z squared plus four. That means to you would equal to z times DZ. And I'm not sure how you were taught to do your substitution to get this into the form that we're trying to match up with. I was just taught to multiply by any Constance inside the integral and then multiply by the reciprocal outside the integral. That was the trick that my calculus teacher taught me way back when and so So I now have a much be not money. 1/2 outside there. Tell you what, Let's clean that up. So we would have are in a girl was the 1/4. And then multiplying by two means I have to modify by 1/2. Outside. There we go. So that gives me 1/8. And now, was he this being a one over you? D you form? That would be the capture log. Uh, the Z squared plus four. And then this next one, we have to realize that this is in the form of one over you squared plus a squared, Do you? We've got a Equalling to in this case because that four has been squared already and new equals z So do you is just that easy. And when we're integrating this form that is one over a times the inverse tangent of you divided by So when we do this last in a girl we end up with 1/2 the one over a day is too times the inverse tangent of are you which is just the letter z divided by a again to And then we would have our arbitrary plus c You could combine your algorithms together, Turn these 1/8 and 1/4 into powers on your arguments and then turn this into a uh actually, I put plus that to be a minus, because over the minus 1/4 let me fix that

So you have this into go in cylindrical corner slinging legal. Yeah. Co ordinates. Uh, no three. So we have been traveling from zero to buy. You drove from, uh, it's your one. I'm from minus 1/2 1 half off. Um, r squared. Sign off. They're square. Plus, he's good. Uh, is she Are you are you hear? Also, um, bounce goods just wanted the sea, so I don't think they were that first. So you were very integrating. Ah, c Q huh? She scored cheese, huh? Did not. And uh huh, See, keep hurts on this one has no factor. So you just integrate stool, see, so that this verse part will be, um while the other two intervals. We mean, they're from jet to buy. I'm just one. And then you have, uh, this function of a week another see? So Oh, this concept, our sign school, huh? Uh, Mrs C of one. Tough. It was one house miners manners on. Have some miners. My as one house. Is she gone into having last one house? So one offers on the house that's going to I've been 40 other term. It will be secure first. So one off Q thirds rhinos. My was one, uh, q thirds. We turn on the miners with their ball comes Oh, there's a minors. So might of my excuses again. A plus. So that this vehicle tow this whole thing, I should be able to two times one. How do you do, sir? Terms 1/3 so that these constables want to obtain peace. Should be on, um, one over two squared times three. Oh, so that is that I'm doing well, then we have the rest of the times already are? Yeah, so that these holy you know, turn seemed to a trail from here to buy. And from 0 to 10 sh are square. Well, no, I'm lying. This is not 1/2 plus 1/2 is not going to school one. So our squirt, sandal filled squared Meyer squirt saying of fate squared, That's one. Well, plus the service that is one of the people too. Plus 1/3 no times. Four Because Daddy's squared from that dance are gr hopes are you are, baby. How Sarah, which is the last thing to go. So now we have to go to the central Are would have begun multiplied these terms and well, first So this internal internal of R D r And then by the time this one, he doesn't have an intro. Well, our cue you are here. So all these one would end up to be with the fall. I took a photo before on these one. It's up to be heart square house So that, um while on the thesis evaluated meeting the end points which are visiting Bala Wait would be uses 1/4 minus zero These where else? We're not going together. This information into living with respect to our we would end up with the angel from zero to buy. Oh, shine square. Well, times, um, the control far Cuba is gonna be applause stands for plus one, huh? Is that so? Plus one, huh? Times three times for bring in that dueted with respect. So all the year you have two new drills? Yes, one for Angel from here to buy off sine squared fair. I'm being or for the Inter American used identity Sinus course about 1/2 one minus. Who signed this? Yeah. Dirty sequel to sign. Well, this is this writer is in quotable Assignment Square said that you wouldn't have these. Inderal on Dana. Well, this internal, these first part would be You're stung, Vera. Hugs because you're going to fly. I'm the general off minus co Sign of my house. The people too minus sign by but like for but these within cereal on to buy. You're so well. These one Where have all the way to the end Points ago China two by zero. Sign off, Sarah. Four by zero. Inside of zero is he sort of sees one we love nothing on what we know so far for there. So what happened? These times are full. So that is the contribution from these first part. The will end up being, uh so to buy. Bye bye to that that he's part B. Why? No, we're not our fine four on this part. The internal order with the just Sarah between cheer on to buy and that ghost of one instead of a view get the difference. All the difference would be dad. When zero turn, these was away and this part would be were by why? Oh, very light. Three times four So that the Holy Grail would be oh, life words plus bye. Working for us or right he's is that I'm being a wheezy sequel to buy over four him. So one bless 1/3 people do 3/3 plus one third, uh, so that these terms is he going to 4/3 very nice. So the four goes away on the Internet should be able to buy hurts.

So you are the poor in job skewed. You go, Porter. It's her and jollies from zero to buy. Right that, monsieur up the world on between our own one were exclude off. War were square Go to Myers are squared. Um, three b c are are Yeah. So these first round's correspond to Z RTR responsive is found on These is obviously for there. So for this first internal awhile the trouble, huh? Three d c is goingto just three times e begin day to bones. So a disabled vehicle toe. So three terms these men see for the upper bound, which is one over square root. Do you mind? Sorry. Squared minus three times you're about We are, um so no shouldn't be of ah is, uh, r between thesis danger next into melted the C on one. Yes, You do by you fear. You know all these internal committee supported into two intervals. Ah, three are over. Screwed of two minus R squared. Rounding drove thio Word. I'm well for dc to go. You can do a use institution. You equal still, um, r squared. Oh, you're close to r squared. You four are squared. Uh, so that these would be, um, these internal three r r through minus R squared one of these, you and then the U legal draw, um, to our pr some of the people to ranger or three over. True because, you know, then you have a physical body are or two, do you? Well, warm nine issue. So the angel off one? Why? Bless you. She told me. Oh, so two times one minus you. Where's that? Um, underwear check is for real friendship. These we differentiate right outside. You should be a day left. So you differentiate. Daddy, have a miner's two on then. Ah, half Because these here today One how? Funding to the minus one, huh? On the minus. Times a minus one. Which you check. You're saying so. This is the r interval on. Uh Well, this is the danger of that. So that bring the three house. So three mine's three hubs and then he squared. Uh, she will, though. Minus free square root of one brain issue on, uh, are you sequel to R squared. So that VC drill they drew up from 01 free R g r. We're square those one. My eyes due to my nose r squared vehicle Thio. So we will do mine. US three. How's this group of Drew miners R squared. But we did meeting zero on one, which he's, um over one. You have a minus. Oh, it was to help. My name's three tens to Mina's. One. Wanted 1/2 on being at your miners, You know, free on then. I am so screwed up too. So that, uh, this will be equal to three grams is for it up too. Mine is. Why? So this is this first thing to go? I mean, you have no danger. A ll three are scored now. The drill three r squared, uh, was that these equal toe and the other guys are cute. So between it's your own one. Yes, you're on. One would be just a kiss between zero and one, which is just one one minus you. So that would be these. There's some of these shooting. Two goes these one when dog one the central bad injury be equal toe. This all these two terms. Three squared off to minors won three times. His queens of Truman is one. Oh, plus one that we're going home are into a lot of these three. Skirt off to minors. One does one. So these waas before enjoyable. Uh, then don't be interpreting zero to play drew fare. That would be just Well, you drink everything. That's it. That's what it is you're applying to be peace You want to buy so so that these internally for Jules three excluded the true minus one loves worms times to buy because, well, we can split the same as this term Times are ready to go, Which is that so they should be. You should see this illusion three times. His quote of two women is one plus one tempted by our prayers free and that this will be on minus three plus one. Did he sound mind still Why?

Um OK, so this problem could be done very, very quickly whether you do a or B, Um, because we're doing the integral from 0 to 2 pi of co sign of Z over the square root of four plus three. Sign of C D. Z. Okay, Uh, because of you, in part B just, you know, just goes from native pi to pie. Um, I'll explain that here because if you let u equal four plus three sign of Z, which would be the method that you would do this substitution. You know, um, de you in this problem would be three co sign of Z DZ. Um, so then, you know, you could you could figure this out, that this is 1/3 to move this over, Miss pieces equal 1/3 to you. This piece is equal to that. But if you know, so if you if you recognize zero and two pi r the same values here in here. So as you plug in and matter of fact, there both zero sign of zero is zero sign of two pies here, so you'll be doing the integral from forward of four of I'll just show you real quick. It's 1/3 view to the negative 1/2 power. Um, I think all my work kind of speaks for itself there. So you could do all this work. But if you're lower and upper bound or the same parents is gonna be zero. And it's really the same work over here for B thing about negative pirates and positive pie in for sign right here are still zero. So you have four plus zero. Um, so part B is gonna really be the same answer because it's the same exact problem. Like a big deal that you change the bounds. Um, but don't waste your time. If if you understand exactly what we have going on, answer is zero for both of them.


Similar Solved Questions

1 answers
2x (3)6 fiz + [dy dx
2x (3)6 fiz + [ dy dx...
5 answers
2. (4 points)} Fir:l the JecobiazLne " trens_o-nati3n
2. (4 points)} Fir:l the Jecobiaz Lne " trens_o-nati3n...
3 answers
Estimate the minimum number of subintervals to approximate the value of3 sin (X + 4)dx with an errorof magnitude less than 2x 10 "4 using a. the error estimate formula for the Trapezoidal Rule_ b. the error estimate formula for Simpson's Rule.
Estimate the minimum number of subintervals to approximate the value of 3 sin (X + 4)dx with an error of magnitude less than 2x 10 "4 using a. the error estimate formula for the Trapezoidal Rule_ b. the error estimate formula for Simpson's Rule....
5 answers
24 Energy _ calculations from wavelength or frequency_ Determine the energy (in ) /photon) of 1023nm lightDetermine the energy (in kJJofa red laser that delivers a 3.56 x1027 photons of 652.7 nm lightA new ultra violet laser deliver 3334.5 kJ of energy using light with a wavelength of 247.6 nm, how many photons of light is needed in this laser pulse?
24 Energy _ calculations from wavelength or frequency_ Determine the energy (in ) /photon) of 1023nm light Determine the energy (in kJJofa red laser that delivers a 3.56 x1027 photons of 652.7 nm light A new ultra violet laser deliver 3334.5 kJ of energy using light with a wavelength of 247.6 nm, ho...
5 answers
(1) Compute the anti-derivative of the following functions: for' 4sin(x) + 2) dx(6) f -8e" dx(c) fk-6sec"() +2-') dc(d) fo-r +2-4/2) dx
(1) Compute the anti-derivative of the following functions: for' 4sin(x) + 2) dx (6) f -8e" dx (c) fk-6sec"() +2-') dc (d) fo-r +2-4/2) dx...
5 answers
(1 point) Write X product X = E Ez Ez of elementary matrices_EjEzC Ez
(1 point) Write X product X = E Ez Ez of elementary matrices_ Ej Ez C Ez...
5 answers
263 362 6k7 | 765 | 12 _ 4=1(c)
263 362 6k7 | 765 | 12 _ 4=1 (c)...
5 answers
@uesdou 11what is the current in a 70-W lightbulb in a socket supplied with 120v?0.58 AS0 A227A17A8400 APrevious
@uesdou 11 what is the current in a 70-W lightbulb in a socket supplied with 120v? 0.58 A S0 A 227A 17A 8400 A Previous...
5 answers
The chapter sections to review are given in parentheses at the end of each question.Indicate if each of the bases in problem 17.57 is found in DNA only, RNA only, or both DNA and RNA. ( 17.1 )
The chapter sections to review are given in parentheses at the end of each question. Indicate if each of the bases in problem 17.57 is found in DNA only, RNA only, or both DNA and RNA. ( 17.1 )...
5 answers
Sum of squares What two nonnegative real numbers $a$ and $b$ whose sum is 23 maximize $a^{2}+b^{2} ?$ Minimize $a^{2}+b^{2} ?$
Sum of squares What two nonnegative real numbers $a$ and $b$ whose sum is 23 maximize $a^{2}+b^{2} ?$ Minimize $a^{2}+b^{2} ?$...
5 answers
The test statistic of z = 1.77 is obtained when testing the claim that p = 1/2 a. Using a significance level of & = 0.10,find the critical value(s). b. Should we reject Ho or should we fail to reject Ho ` Clickhere to_view_page of the standard normal distribution table Click here to_view_page 2 of the standard normal distribution tablea. The critical value(s) islare z = (Round to two decimal places as needed. Use a comma to separate answers as needed:)
The test statistic of z = 1.77 is obtained when testing the claim that p = 1/2 a. Using a significance level of & = 0.10,find the critical value(s). b. Should we reject Ho or should we fail to reject Ho ` Clickhere to_view_page of the standard normal distribution table Click here to_view_page 2 ...
5 answers
Chapter 4, Section 4.6, Supplementary Question 01Find the coordinate vector for p relative to S = {p1 pz pz} p = 5 - Sx + x2; Pi = 1, Pz = X, P3 = x2 [pJs
Chapter 4, Section 4.6, Supplementary Question 01 Find the coordinate vector for p relative to S = {p1 pz pz} p = 5 - Sx + x2; Pi = 1, Pz = X, P3 = x2 [pJs...
5 answers
W 04 ! W 4 8 WN [ V ' 1 6 W 6 J04 WL W
W 04 ! W 4 8 WN [ V ' 1 6 W 6 J04 WL W...
5 answers
SaveExit Certify Lesson: 4 1 Relations and FunctonsQOuesrionof 9, stCpconrectDetermine g6 -R(x) for the following Fundors() = 42AnsweE( + 4) RG)Gro7oHaute Lcamnine
Save Exit Certify Lesson: 4 1 Relations and Functons QOuesrion of 9, stCp conrect Determine g6 - R(x) for the following Fundor s() = 42 Answe E( + 4) RG) Gro7oHaute Lcamnine...
1 answers
Consider the linear transformation T(x, Y, 2) = (3x + Y + Sz,x _ 2y+ 22,(4 _ a)y) where a is a constant and a # 4a) [2 marks] Show that T is linear by finding the standard matrix associated with Tb) [2 marks] Find the image of (~3,0,3) under T .c) [1 marks] Let A be the standard matrix associated with T Prove that A is invertible then find A-1_[3 marks] Find the pre-image of (5,3,0) under T _
Consider the linear transformation T(x, Y, 2) = (3x + Y + Sz,x _ 2y+ 22,(4 _ a)y) where a is a constant and a # 4 a) [2 marks] Show that T is linear by finding the standard matrix associated with T b) [2 marks] Find the image of (~3,0,3) under T . c) [1 marks] Let A be the standard matrix associated...
5 answers
The ext few problems uses the function f : (0,1) 4 (0,1) that we HOW define: If a € (0,1). let (k;)i, be decimal expansion for with the choice of the one ending in (000o if a € E, define f (a) CiIT5- (So; for example f(0.123) 0.010203.What is f(1/3) as a fraction? 6. Show that f is continous at a if and only if a € E. (Hint: estimate |f (a) _ f(6)| if a.b have the same first digits in their decimal expansion_
The ext few problems uses the function f : (0,1) 4 (0,1) that we HOW define: If a € (0,1). let (k;)i, be decimal expansion for with the choice of the one ending in (000o if a € E, define f (a) CiIT5- (So; for example f(0.123) 0.010203. What is f(1/3) as a fraction? 6. Show that f is con...
5 answers
Suppose fn 7f uniformly on set S and gn 9 uniformly on S. Recall the definition that fn 7f uniformly on S. (You do not need to write this down: This only serves as a hint for next parts:) (b) Use the definition in a) to show that fn + In - f + g uniformly on S.(c) Suppose h S _ R is a bounded function. Is fnh 7 fh uniformly on S? Justify your answer_
Suppose fn 7f uniformly on set S and gn 9 uniformly on S. Recall the definition that fn 7f uniformly on S. (You do not need to write this down: This only serves as a hint for next parts:) (b) Use the definition in a) to show that fn + In - f + g uniformly on S. (c) Suppose h S _ R is a bounded funct...

-- 0.019105--