MATH 381 HOMEWORK 2 SOLUTIONS


 Ashley Miller
 1 years ago
 Views:
Transcription
1 MATH 38 HOMEWORK SOLUTIONS Question (p.86 #8). If g(x)[e y e y ] is harmonic, g() =,g () =, find g(x). Let f(x, y) = g(x)[e y e y ].Then Since f(x, y) is harmonic, f + f = and we require x y f x = g (x)[e y e y ] f y = 4g(x)[ey e y ]. g (x) + 4g(x) =. Thus g(x) has the form A sin(x) + B cos(x) and by the initial conditions, A = andb =. Therefore, g(x) = sin(x). Question (p.86 #). Find the harmonic conjugate of tan x y where π < tan x y π. Write u(x, y) = tan x y.thenbythecauchyriemannequations, () () By (), x = y x + y y = y x + y = v y y = y x x + y y = x x + y = v x. v = log(x + y ) + C(x), and by () v x = x x + y + x C (x) = x + y so C (x) = andc(x) is a constant, call it D. Therefore, v(x, y) = log(x + y ) + D. Question 3. (p.86 #3) Show, if u(x, y) and v(x, y) are harmonic functions, that u + v must be a harmonic function but that uv need not be a harmonic function. Is e u e v a harmonic function? If u and v are harmonic, then u + v is harmonic since (u + v) + (u + v) = u x y x + v x + u y + v y = u x + u y + v x + v y =. To show that uv is not necessarily harmonic, it suffices to show that there exists u, v harmonic such that (uv) x + (uv) y = v x x + v y y. Any u = v harmonic where, willsuffice. Forinstance,takingu = v = x will work, since it s harmonic (both x y of its secondorder partials vanish) but v x x + v y y =. Date: October3,.
2 MATH 38 HOMEWORK SOLUTIONS Now, in order for e u e v to be harmonic, we need (e u e v ) + (e u e v ) = e u+v x y x + v x + y + v y =. Thus, the existence of any u, v harmonic such that + v x x + + v y y willshowthate u e v is not harmonic. Again, taking u = v = x gives us what we want as e x is easily seen to be nonharmonic. Question 4 (p.6 #4). State the domain of analyticity of f(z) = e iz. Find the real and imaginary parts u(x, y) and v(x, y) of the function, show that these satisfy the CauchyRiemann equations, and find f (z) in terms of z. By definition, Therefore, These are continuous functions at all (x, y) R.Now, f(z) = e iz = e ix e y = e y [cos x + i sin x]. u(x, y) = e y cos x v(x, y) = e y sin x. x = e y sin x = v y y = e y cos x = v x so u, v satisfy the CR equations, and these derivatives are continuous for allx, y. Therefore, f(z) is entire. Furthermore, f (z) = e y sin x + i(e y cos x) = i(e y [cos x + i sin x]) = ie iz. Question 5 (p.6 #6). State the domain of analyticity of f(z) = e ez. Find the real and imaginary parts u(x, y) and v(x, y) of the function, show that these satisfy the CauchyRiemann equations, and find f (z) in terms of z. First, observe that f is an entire function of an entire function, so it is analytic everywhere. Now, so e ez = e ex (cos y+i sin y) = e ex cos y cos(e x sin y) + i sin(e x sin y), u(x, y) = e ex cos y cos(e x sin y) v(x, y) = e ex cos y sin(e x sin y) and f satisfies the CR equations. Furthermore, x = x cos y ee (e x cos y) cos(e x sin y) e ex cos y (e x sin y) sin(e x sin y) = e ex cos y+x (cos y cos(e x sin y) sin y sin(e x sin y)) v y = x cos y ee ( e x sin y) sin(e x sin y) + e ex cos y cos(e x sin y)(e x cos y) = e ex cos y+x (cos y cos(e x sin y) sin y sin(e x sin y)) y = x cos y ee ( e x sin y) cos(e x sin y) + e ex cos y (e x cos y)( sin(e x sin y)) = e ex cos y+x (cos y sin(e x sin y) + sin y cos(e x sin y)) v x = x cos y ee (e x cos y) sin(e x sin y) + e ex cos y (e x sin y) cos(e x sin y) = e ex cos y+x (cos y sin(e x sin y) + sin y cos(e x sin y)) f (z) = e ex cos y e x (cos y cos(e x sin y) sin y sin(e x sin y)) + i(cos y sin(e x sin y) + sin y cos(e x sin y) = e ex cos y e x cos y cos(e x sin y) + i sin(e x sin y) + e x sin yi cos(e x sin y) sin(e x sin y) = e ex cos y cos(e x sin y) + i sin(e x sin y)e x (cos y + i sin y) = e ez e z.
3 MATH 38 HOMEWORK SOLUTIONS 3 Question 6 (p.6 #3). (a) Prove the expression given in the text for the n th derivative of f(t) = t t + = Re. (Note: t R). t i (b) Find similar expressions for the n th derivative of f(t) = t + = Im.(Note: t R ). t i (a) By the Lemma, for n, Now, observe that f (n) (t) = Re dn dt n t i = Re ( )n n! (t i) n+ = t+i, so by the binomial theorem t i t + ( ) n n! (t i) = ( )n n!(t + i) n+ = ( )n n+ n!(n + )! i k t n+ k n+ (t + ) n+ (t + ) n+ k= (n + k)!k!. But notice that we only get contributions to the real part of this expression when k is even; i.e. when i k R. Summing over the even integers, k = m, we get for n odd that and for n even that (b) In this case we want f (n) (t) = ( )n n!(n + )! (t + ) n+ = f (n) (t) = ( )n!(n + )! (t + ) n+ n!(n + )! (t + ) n+ n+ m= n+ m= n m= i m t n+ m (n + m)!(m)! ( ) m t n+ m (n + m)!(m)! i m t n+ m (n + m)!(m)! f (n) (t) = Im dn dt n t i = Im ( )n n! (t i) n+. By the work above, we want the imaginary part of ( ) n n!(n + )! (t + ) n+ n+ k= i k t n+ k (n + k)!k!. In this case we get contributions when k is odd, so we take the the sum over k = m + for m. Note that i m+ = ( ) m i. It follows that when n is odd, and when n is even, f (n) (t) = f (n) (t) = ( )n!(n + )! (t + ) n+ n m= n n!(n + )! (t + ) n+ m= ( ) m t n m (n m)!(m + )! ( ) m t n m (n m)!(m + )!. Question 7 (p.6 #5). Let P (ψ) = N n= e inψ. (a) Show that (b) Find lim ψ P (ψ). (c) Plot P (ψ) for ψ and N = 3. (a) Note that P (ψ) = sin(nψ) sin(ψ). P (ψ) = einψ e iψ.
4 z = π 4 + kπ, k Z. 4 MATH 38 HOMEWORK SOLUTIONS Thus, Thus, P (ψ) = einψ e iψ = einψ e iψ = einψ e iψ = einψ e iψ (b) By l Hopital s rule we get einψ e inψ e iψ e iψ cos(nψ) + i sin(nψ) cos( Nψ) i sin( Nψ) cos(ψ) + i sin(ψ) cos( ψ) i sin( ψ) i sin(nψ) i sin ψ. P (ψ) = einψ e iψ sin(nψ) sin ψ = sin(nψ) sin ψ. sin(nψ) Nsin(Nψ) lim = lim = N. ψ sin ψ ψ sinψ (c) If you have nothing else, just plug it in Wolfram Alpha. Question 8 (p. #7). Show that sin z cos z = has solutions only for real values of z. What are the solutions? In other words, for z = x + iy we want Equating the real parts and imaginary parts we require (3) (4) sin x cosh y + i cos x sinh y = cos x cosh y i sin x sinh y. sin x cosh y = cos x cosh y cos x sinh y = sin x sinh y. Suppose y andhencesinhy andcoshy. Then in order to have solutions, by (3), weneedcosx = sin x and by (4) we need cos x = sin x. These equations are only satisfied for sin x = cos x =, but no solutions for x exists. Therefore, if there are solutions to the original equation, we must have that y =. Suppose y =. Then since cosh = andsinh= wesimplyneedsolutionstosinx = cos x. Thus we have solutions if and only if Question 9 (p. #). Where does the function f(z) = 3sinz cos z fail to be analytic? Since sin z and cos z are both analytic, f(z) will fail to be analytic when 3sinz cos z =. In other words, when we have solutions to ( sin x cosh y + i cos x sinh y) = cos x cosh y i sin x sinh y. Equating the real parts and imaginary parts we require (5) 3sinx cosh y = cos x cosh y (6) 3cosx sinh y = sin x sinh y. By the same argument as the previous question, there are no solutions when y. Suppose y =. Then since cosh = andsinh= wesimplyneedsolutionsto 3sinx = cos x, thatistotanx = 3. So f(z) is not analytic when z = π 6 + kπ, k Z. Question (p. #). Let f(z) = sin z. (a) Express this function in the form u(x, y) + iv(x, y). Where in the complex plane is this function analytic? (b) What is the derivative of f(z)? Where in the complex plane is f (z) analytic?
5 MATH 38 HOMEWORK SOLUTIONS 5 (a) Since sin z is entire, and z (b) For z, is analytic for z, it follows that f(z) is analytic for z. sin x iy = sin z x + y x = sin x + y cosh y x + y + i cos x x + y sinh y x + y x = sin x + y cosh y x + y i cos x x + y sinh y x + y. which is analytic for all z. d dz sin z = cos z z Question (p. #5). Show that cos z = sinh y + cos x. cos z = cos x cosh y i sin x sinh y = cos x cosh y + sin x sinh y = cos x( + sinh y) + sin x sinh y = cos x + sinh y(cos x + sin x) = cos x + sinh y Question (p.9 #6). Use logarithms to find solutions to e z = e iz. We want solutions to e z( i) =, so taking logs on both sides we get for any k Z, z( i) = πik, so z = πik (i + )πik = = (i )kπ. i Question 3 (p.9 #8). Use logarithms to find solutions to e z = (e z ). In other words, we want solutions to e z 3e z + =. By the quadratic formula, we get that Taking logs gives that for k Z. e z = 3 ± 4 9 = 3 ± 5. z = log 3 ± 5 + πik Question 4 (p.9 #). Use logarithms to find solutions to e ez =. First, taking logs we get e z = πik for k Z. Now for k >, the argument of πik is π + πm where m Z, and for k <, the argument of πik is 3π + πm (again m Z). Thus, for k >, z = log(πk) + i π + mπ and for k <, Question 5 (p.9 #3). Show that where θ R and e iθ. z = log(πk) + i π + mπ. Re log( + e iθ ) = log cos θ
6 6 MATH 38 HOMEWORK SOLUTIONS Re log( + e iθ ) = log + e iθ = log ( + cos θ) + sin θ = log( + cosθ) = log cos θ + sin θ + cos θ sin θ = log cos θ Question 6 (p.7 #9). Intergrate along z =, in the lower half plane. z dz Let z = e it, then we are integrating along the interval t [, π]. Now,dz = ie it dt so z dz = π e i t iei tdt = iπ. Question 7 (p.7 #). Show that x = cos t, y = sin t, wheret ranges from to π, yields a parametric representation of the ellipse x 4 + y =. Use this representation to evaluate i zdz along the portion of the ellipse in the first quadrant. Note that (cost) + sin t = cos t + sin t = 4 and furthermore cos = cosπ = andsin= sin π =. To see that we get all of the ellipse, note that x = cost has solutions t [, π] for all x [, ] and y = sin t has solutions t [, π] for all y [, ]. Furthermore, the parametrization is : except for when x =,y =. Setting z = x + iy = cost + i sin t, wegetdz = (i cos t sint)dt, and i zdz = = π π = 3 + iπ. (cost i sin t)(i cos t sint)dt (i 3sint cos t)dt Question 8 (p.7 #4). Consider I = +i e z dz taken along the line x = y. Without actually doing the integration, show that I 5e 3. Let M be the maximal value attained by e z along the path of integration. Now, for x = y, e z = e x y +ixy = e 3y which attains its maximum when y attains a maximum that is, when z = +i. Therefore M = e 3. By the pythagorean theorem, the length of the path is + = 5, so by the ML inequality, I ML = 5e 3. Question 9 (p.7 #6). Consider I = i e i log z dz taken along the parabola y = x. Without doing the integration, show that I.479e π. Letting θ = arg z e i log z = e i(logz iθ) = e i logz e θ = e θ. Along the given path, this attains a maximum when θ = π, so let M = e π.
7 MATH 38 HOMEWORK SOLUTIONS 7 Now, we need to find the length of the path of integration. So since dy = xdx, L = + dy dx dx = + 4x dx The ML inequality then gives the desired result. <.479. Question (p.8 #). Is the CauchyGoursat theorem directly applicable to z= sin z z+i dz? Since sin z z+i directly applicable. is analytic everywhere except for z = i which is not in the unit circle, the CG theorem is Question (p.8 #6). Is the CauchyGoursat theorem directly applicable to z i = log zdz? Since is not in the unit circle about i +, log z is analytic in the desired region so the CG theorem is directly applicable. Question (p.8 #7). Is the CauchyGoursat theorem directly applicable to z= (z ) 4 + dz? Observe that we have a singularity when z is a primitive 8 th root of unity that is, when (z ) 4 =. These roots of unity lie on the unit circle, so shifting over by, we need to determine if the roots closest to the origin, at z = e iπ34 + andz = e iπ54 + have absolute value greater than. By geometry (right angle triangles), it can be seen that at these points, so the CG theorem directly applies. z = + > Question 3 (p.8 #9). Is the CauchyGoursat theorem directly applicable to z=b z + bz + dz where < b <? In this case the singularities are at the roots of the equation x + bx +, that is, when z = b ± b 4 = b 4 b ± i. Here, b z = b = > b 4 therefore CG applies directly. Question 4 (p.8 #3). Prove that π e cos θ sin(sin θ + θ)dθ =. Begin with e z dz performed around z =. Use the parametric representation z = e iθ, θ π. Separate your equation into real and imaginary parts. Let z = e iθ = cos θ + i sin θ, sodz = e iθ idθ. Sincee z is analytic, But then π z= e z dz = e cos θ+i(sin θ+θ) dθ = π e cos θ+i sin θ e iθ idθ =. so by equating the imaginary part with zero we get the desired result. π e cos θ (cos(θ + sin θ) + i sin(θ + sin θ))dθ =
Solutions for Math 311 Assignment #1
Solutions for Math 311 Assignment #1 (1) Show that (a) Re(iz) Im(z); (b) Im(iz) Re(z). Proof. Let z x + yi with x Re(z) and y Im(z). Then Re(iz) Re( y + xi) y Im(z) and Im(iz) Im( y + xi) x Re(z). () Verify
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationMath Spring 2014 Solutions to Assignment # 4 Completion Date: Friday May 16, f(z) = 3x + y + i (3y x)
Math 311  Spring 2014 Solutions to Assignment # 4 Completion Date: Friday May 16, 2014 Question 1. [p 77, #1 (a)] Apply the theorem in Sec. 22 to verify that the function is entire. f(z) = 3x + y + i
More informationSolutions to Practice Final Exam
Math V22. Calculus IV, ection, pring 27 olutions to Practice Final Exam Problem Consider the integral x 2 2x dy dx + 2x dy dx x 2 x (a) ketch the region of integration. olution: ee Figure. y (2, ) y =
More information2 Complex Functions and the CauchyRiemann Equations
2 Complex Functions and the CauchyRiemann Equations 2.1 Complex functions In onevariable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationAgain, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions.
Chapter 4 Complex Analysis 4.1 Complex Differentiation Recall the definition of differentiation for a real function f(x): f f(x + δx) f(x) (x) = lim. δx 0 δx In this definition, it is important that the
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More information4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complexvalued function of a real variable
4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complexvalued function of a real variable Consider a complex valued function f(t) of a real variable
More information6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.
hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,
More informationComplex Algebra. What is the identity, the number such that it times any number leaves that number alone?
Complex Algebra When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in some sense not real. Later, when probably one of the students of Pythagoras
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More informationTHE COMPLEX EXPONENTIAL FUNCTION
Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula
More informationCOMPLEX NUMBERS AND SERIES. Contents
COMPLEX NUMBERS AND SERIES MIKE BOYLE Contents 1. Complex Numbers Definition 1.1. A complex number is a number z of the form z = x + iy, where x and y are real numbers, and i is another number such that
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationPractice Problems for Midterm 2
Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationCalculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationEuler s Formula Math 220
Euler s Formula Math 0 last change: Sept 3, 05 Complex numbers A complex number is an expression of the form x+iy where x and y are real numbers and i is the imaginary square root of. For example, + 3i
More information5 Indefinite integral
5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse
More informationFrom the first principles, we define the complex exponential function as a complex function f(z) that satisfies the following defining properties:
3. Exponential and trigonometric functions From the first principles, we define the complex exponential function as a complex function f(z) that satisfies the following defining properties: 1. f(z) is
More informationC. Complex Numbers. 1. Complex arithmetic.
C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
More informationChapter 4. Linear Second Order Equations. ay + by + cy = 0, (1) where a, b, c are constants. The associated auxiliary equation is., r 2 = b b 2 4ac 2a
Chapter 4. Linear Second Order Equations ay + by + cy = 0, (1) where a, b, c are constants. ar 2 + br + c = 0. (2) Consequently, y = e rx is a solution to (1) if an only if r satisfies (2). So, the equation
More informationSept 20, 2011 MATH 140: Calculus I Tutorial 2. ln(x 2 1) = 3 x 2 1 = e 3 x = e 3 + 1
Sept, MATH 4: Calculus I Tutorial Solving Quadratics, Dividing Polynomials Problem Solve for x: ln(x ) =. ln(x ) = x = e x = e + Problem Solve for x: e x e x + =. Let y = e x. Then we have a quadratic
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationMMGF30, Transformteori och analytiska funktioner
MATEMATIK Göteborgs universitet Tentamen 06038, 8:30:30 MMGF30, Transformteori och analytiska funktioner Examiner: Mahmood Alaghmandan, tel: 77 53 74, Email: mahala@chalmers.se Telefonvakt: 07 97 5630
More informationLecture 17: Conformal Invariance
Lecture 17: Conformal Invariance Scribe: Yee Lok Wong Department of Mathematics, MIT November 7, 006 1 Eventual Hitting Probability In previous lectures, we studied the following PDE for ρ(x, t x 0 ) that
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationChapter 2. Complex Analysis. 2.1 Analytic functions. 2.1.1 The complex plane
Chapter 2 Complex Analysis In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _i FIGURE Complex numbers as points in the Arg plane i _i +i i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationReview Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
More informationDEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x
Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationDIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents
DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the CauchyRiemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition
More informationSECONDORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
L SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A secondorder linear differential equation is one of the form d
More informationMATH : HONORS CALCULUS3 HOMEWORK 6: SOLUTIONS
MATH 1630033: HONORS CALCULUS3 HOMEWORK 6: SOLUTIONS 251 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have
More informationInverse Circular Function and Trigonometric Equation
Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationRecall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:
Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1
More informationExample 1. Example 1 Plot the points whose polar coordinates are given by
Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationSolutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationSolutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in
KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set
More informationElementary Functions
Chapter Three Elementary Functions 31 Introduction Complex functions are, of course, quite easy to come by they are simply ordered pairs of realvalued functions of two variables We have, however, already
More informationIntermediate Value Theorem, Rolle s Theorem and Mean Value Theorem
Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem February 21, 214 In many problems, you are asked to show that something exists, but are not required to give a specific example or formula
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
More information7. Cauchy s integral theorem and Cauchy s integral formula
7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose
More informationMath 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).
Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationSolutions to Homework 5
Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More information11.7 Polar Form of Complex Numbers
11.7 Polar Form of Complex Numbers 989 11.7 Polar Form of Complex Numbers In this section, we return to our study of complex numbers which were first introduced in Section.. Recall that a complex number
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationFunction Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015
Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax
More informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More informationMerton College Maths for Physics Prelims October 10, 2005 MT I. Calculus. { y(x + δx) y(x)
Merton College Maths for Physics Prelims October 10, 2005 1. From the definition of the derivative, dy = lim δx 0 MT I Calculus { y(x + δx) y(x) evaluate d(x 2 )/. In the same way evaluate d(sin x)/. 2.
More information2. Functions and limits: Analyticity and Harmonic Functions
2. Functions and limits: Analyticity and Harmonic Functions Let S be a set of complex numbers in the complex plane. For every point z = x + iy S, we specific the rule to assign a corresponding complex
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationEXAM. Practice Questions for Exam #2. Math 3350, Spring April 3, 2004 ANSWERS
EXAM Practice Questions for Exam #2 Math 3350, Spring 2004 April 3, 2004 ANSWERS i Problem 1. Find the general solution. A. D 3 (D 2)(D 3) 2 y = 0. The characteristic polynomial is λ 3 (λ 2)(λ 3) 2. Thus,
More informationHow to add sine functions of different amplitude and phase
Physics 5B Winter 2009 How to add sine functions of different amplitude and phase In these notes I will show you how to add two sinusoidal waves each of different amplitude and phase to get a third sinusoidal
More informationInformal lecture notes for complex analysis
Informal lecture notes for complex analysis Robert Neel I ll assume you re familiar with the review of complex numbers and their algebra as contained in Appendix G of Stewart s book, so we ll pick up where
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationCalculus with Analytic Geometry I Exam 10 Take Home part
Calculus with Analytic Geometry I Exam 10 Take Home part Textbook, Section 47, Exercises #22, 30, 32, 38, 48, 56, 70, 76 1 # 22) Find, correct to two decimal places, the coordinates of the point on the
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationMath 201 Lecture 23: Power Series Method for Equations with Polynomial
Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in
More informationComputing Euler angles from a rotation matrix
Computing Euler angles from a rotation matrix Gregory G. Slabaugh Abstract This document discusses a simple technique to find all possible Euler angles from a rotation matrix. Determination of Euler angles
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationis not a real number it follows that there is no REAL
210 CHAPTER EIGHT 8. Complex Numbers When we solve x 2 + 2x + 2 = 0 and use the Quadratic Formula we get Since we know that solution to the equation x 2 + 2x + 2 = 0. is not a real number it follows that
More informationMath 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y
Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve at the point (0, 0) is cosh y = x + sin y + cos y Answer : y = x Justification: The equation of the
More informationAlgebra 2: Themes for the Big Final Exam
Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationChapter 9. Miscellaneous Applications
hapter 9 53 Miscellaneous Applications In this chapter we consider selected methods where complex variable techniques can be employed to solve various types of problems In particular, we consider the summation
More informationFirst Order NonLinear Equations
First Order NonLinear Equations We will briefly consider nonlinear equations. In general, these may be much more difficult to solve than linear equations, but in some cases we will still be able to solve
More informationHomework #1 Solutions
MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From
More information14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style
Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationThe Math Circle, Spring 2004
The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is NonEuclidean Geometry? Most geometries on the plane R 2 are noneuclidean. Let s denote arc length. Then Euclidean geometry arises from the
More informationExamples of Functions
Chapter 3 Examples of Functions Obvious is the most dangerous word in mathematics. E. T. Bell 3.1 Möbius Transformations The first class of functions that we will discuss in some detail are built from
More informationHIGHER ORDER DIFFERENTIAL EQUATIONS
HIGHER ORDER DIFFERENTIAL EQUATIONS 1 Higher Order Equations Consider the differential equation (1) y (n) (x) f(x, y(x), y (x),, y (n 1) (x)) 11 The Existence and Uniqueness Theorem 12 The general solution
More informationNotes and questions to aid Alevel Mathematics revision
Notes and questions to aid Alevel Mathematics revision Robert Bowles University College London October 4, 5 Introduction Introduction There are some students who find the first year s study at UCL and
More informationtegrals as General & Particular Solutions
tegrals as General & Particular Solutions dy dx = f(x) General Solution: y(x) = f(x) dx + C Particular Solution: dy dx = f(x), y(x 0) = y 0 Examples: 1) dy dx = (x 2)2 ;y(2) = 1; 2) dy ;y(0) = 0; 3) dx
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationCOMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS
COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS BORIS HASSELBLATT CONTENTS. Introduction. Why complex numbers were introduced 3. Complex numbers, Euler s formula 3 4. Homogeneous differential equations 8 5.
More informationbe a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that
Problem 1A. Let... X 2 X 1 be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that i=1 X i is nonempty and connected. Since X i is closed in X, it is compact.
More informationCALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.
CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n
More information