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Lecture notes in Differential Equations

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Introduction to Differential Equations Lecture notes for MATH 2351/2352 Jeffrey R. Chasnov k K k m m x x 1 2 The Hong Kong University of Science and TechnologyThe Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon Hong Kong Copyright○ c 2009–2016 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.Preface What follows are my lecture notes for a first course in differential equations, taught at the Hong Kong University of Science and Technology. Included in these notes are links to short tutorial videos posted on YouTube. Much of the material of Chapters 2-6 and 8 has been adapted from the widely used textbook “Elementary differential equations and boundary value problems” c by Boyce & DiPrima (John Wiley & Sons, Inc., Seventh Edition,○ 2001). Many of the examples presented in these notes may be found in this book. The material of Chapter 7 is adapted from the textbook “Nonlinear dynamics and chaos” by Steven c H. Strogatz (Perseus Publishing,○ 1994). All web surfers are welcome to download these notes, watch the YouTube videos, and to use the notes and videos freely for teaching and learning. An associated free review book with links to YouTube videos is also available from the ebook publisher bookboon.com. I welcome any comments, suggestions or corrections sent by email to jeffrey.chasnovust.hk. Links to my website, these lecture notes, my YouTube page, and the free ebook from bookboon.com are given below. Homepage: http://www.math.ust.hk/machas YouTube: https://www.youtube.com/user/jchasnov Lecture notes: http://www.math.ust.hk/machas/differential-equations.pdf Bookboon: http://bookboon.com/en/differential-equations-with-youtube-examples-ebook iiiContents 0 A short mathematical review 1 0.1 The trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 The exponential function and the natural logarithm . . . . . . . . . . . 1 0.3 Definition of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4 Differentiating a combination of functions . . . . . . . . . . . . . . . . 2 0.4.1 The sum or difference rule . . . . . . . . . . . . . . . . . . . . . 2 0.4.2 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4.3 The quotient rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.5 Differentiating elementary functions . . . . . . . . . . . . . . . . . . . . 3 0.5.1 The power rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.5.2 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . 3 0.5.3 Exponential and natural logarithm functions . . . . . . . . . . . 3 0.6 Definition of the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.7 The fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . 4 0.8 Definite and indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . 5 0.9 Indefinite integrals of elementary functions . . . . . . . . . . . . . . . . 5 0.10 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.11 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.12 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.13 Functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . 7 0.14 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Introduction to odes 13 1.1 The simplest type of differential equation . . . . . . . . . . . . . . . . . 13 2 First-order odes 15 2.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Separable equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Compound interest . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 Chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.3 Terminal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.4 Escape velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.5 RC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.6 The logistic equation . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 Second-order odes, constant coefficients 31 3.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 The principle of superposition . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Homogeneous odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.1 Real, distinct roots . . . . . . . . . . . . . . . . . . . . . . . . . . 34 vCONTENTS 3.4.2 Complex conjugate, distinct roots . . . . . . . . . . . . . . . . . 36 3.4.3 Repeated roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Inhomogeneous odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 First-order linear inhomogeneous odes revisited . . . . . . . . . . . . . 42 3.7 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.8 Damped resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4 The Laplace transform 49 4.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Solution of initial value problems . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Heaviside and Dirac delta functions . . . . . . . . . . . . . . . . . . . . 55 4.3.1 Heaviside function . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.2 Dirac delta function . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Discontinuous or impulsive terms . . . . . . . . . . . . . . . . . . . . . 59 5 Series solutions 63 5.1 Ordinary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Regular singular points: Cauchy-Euler equations . . . . . . . . . . . . 66 5.2.1 Real, distinct roots . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2.2 Complex conjugate roots . . . . . . . . . . . . . . . . . . . . . . 69 5.2.3 Repeated roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6 Systems of equations 71 6.1 Matrices, determinants and the eigenvalue problem . . . . . . . . . . . 71 6.2 Coupled first-order equations . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2.1 Two distinct real eigenvalues . . . . . . . . . . . . . . . . . . . . 74 6.2.2 Complex conjugate eigenvalues . . . . . . . . . . . . . . . . . . 78 6.2.3 Repeated eigenvalues with one eigenvector . . . . . . . . . . . 79 6.3 Normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7 Nonlinear differential equations 85 7.1 Fixed points and stability . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.1.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.1.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2 One-dimensional bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2.1 Saddle-node bifurcation . . . . . . . . . . . . . . . . . . . . . . . 89 7.2.2 Transcritical bifurcation . . . . . . . . . . . . . . . . . . . . . . . 90 7.2.3 Supercritical pitchfork bifurcation . . . . . . . . . . . . . . . . . 91 7.2.4 Subcritical pitchfork bifurcation . . . . . . . . . . . . . . . . . . 92 7.2.5 Application: a mathematical model of a fishery . . . . . . . . . 94 7.3 Two-dimensional bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 95 7.3.1 Supercritical Hopf bifurcation . . . . . . . . . . . . . . . . . . . 96 7.3.2 Subcritical Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . 97 8 Partial differential equations 99 8.1 Derivation of the diffusion equation . . . . . . . . . . . . . . . . . . . . 99 8.2 Derivation of the wave equation . . . . . . . . . . . . . . . . . . . . . . 100 8.3 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.4 Fourier cosine and sine series . . . . . . . . . . . . . . . . . . . . . . . . 103 8.5 Solution of the diffusion equation . . . . . . . . . . . . . . . . . . . . . 106 8.5.1 Homogeneous boundary conditions . . . . . . . . . . . . . . . . 106 8.5.2 Inhomogeneous boundary conditions . . . . . . . . . . . . . . . 110 vi CONTENTSCONTENTS 8.5.3 Pipe with closed ends . . . . . . . . . . . . . . . . . . . . . . . . 111 8.6 Solution of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . 113 8.6.1 Plucked string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.6.2 Hammered string . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.6.3 General initial conditions . . . . . . . . . . . . . . . . . . . . . . 115 8.7 The Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.7.1 Dirichlet problem for a rectangle . . . . . . . . . . . . . . . . . . 116 8.7.2 Dirichlet problem for a circle . . . . . . . . . . . . . . . . . . . . 118 8.8 The Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.8.1 Heuristic derivation of the Schrödinger equation . . . . . . . . 121 8.8.2 The time-independent Schrödinger equation . . . . . . . . . . . 123 8.8.3 Particle in a one-dimensional box . . . . . . . . . . . . . . . . . 123 8.8.4 The simple harmonic oscillator . . . . . . . . . . . . . . . . . . . 124 8.8.5 Particle in a three-dimensional box . . . . . . . . . . . . . . . . 127 8.8.6 The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . 128 CONTENTS viiCONTENTS viii CONTENTSChapter 0 A short mathematical review A basic understanding of calculus is required to undertake a study of differential equations. This zero chapter presents a short review. 0.1 The trigonometric functions The Pythagorean trigonometric identity is 2 2 sin x+ cos x = 1, and the addition theorems are sin(x+ y)= sin(x) cos(y)+ cos(x) sin(y), cos(x+ y)= cos(x) cos(y)− sin(x) sin(y). Also, the values of sin x in the first quadrant can be remembered by the rule of ∘ ∘ ∘ ∘ ∘ quarters, with 0 = 0, 30 = p/6, 45 = p/4, 60 = p/3, 90 = p/2: r r r 0 1 2 ∘ ∘ ∘ sin 0 = , sin 30 = , sin 45 = , 4 4 4 r r 3 4 ∘ ∘ sin 60 = , sin 90 = . 4 4 The following symmetry properties are also useful: sin(p/2− x)= cos x, cos(p/2− x)= sin x; and sin(− x)=− sin(x), cos(− x)= cos(x). 0.2 The exponential function and the natural logarithm The transcendental number e, approximately 2.71828, is defined as   n 1 e= lim 1+ . n→¥ n x The exponential function exp(x) = e and natural logarithm ln x are inverse func- tions satisfying ln x x e = x, ln e = x. The usual rules of exponents apply: x y x+y x y x− y x p px e e = e , e /e = e , (e ) = e . The corresponding rules for the logarithmic function are p ln(xy)= ln x+ ln y, ln(x/y)= ln x− ln y, ln x = p ln x. 10.3. DEFINITION OF THE DERIVATIVE 0.3 Definition of the derivative ′ The derivative of the function y = f(x), denoted as f (x) or dy/dx, is defined as the slope of the tangent line to the curve y = f(x) at the point (x, y). This slope is obtained by a limit, and is defined as f(x+ h)− f(x) ′ f (x)= lim . (1) h h→0 0.4 Differentiating a combination of functions 0.4.1 The sum or difference rule The derivative of the sum of f(x) and g(x) is ′ ′ ′ ( f + g) = f + g . Similarly, the derivative of the difference is ′ ′ ′ ( f− g) = f − g . 0.4.2 The product rule The derivative of the product of f(x) and g(x) is ′ ′ ′ ( f g) = f g+ f g , and should be memorized as “the derivative of the first times the second plus the first times the derivative of the second.” 0.4.3 The quotient rule The derivative of the quotient of f(x) and g(x) is   ′ ′ ′ f f g− f g = , 2 g g and should be memorized as “the derivative of the top times the bottom minus the top times the derivative of the bottom over the bottom squared.” 0.4.4 The chain rule The derivative of the composition of f(x) and g(x) is   ′ ′ ′ f(g(x)) = f (g(x))· g (x), and should be memorized as “the derivative of the outside times the derivative of the inside.” 2 CHAPTER 0. A SHORT MATHEMATICAL REVIEW0.5. DIFFERENTIATING ELEMENTARY FUNCTIONS 0.5 Differentiating elementary functions 0.5.1 The power rule The derivative of a power of x is given by d p p− 1 x = px . dx 0.5.2 Trigonometric functions The derivatives of sin x and cos x are ′ ′ (sin x) = cos x, (cos x) =− sin x. We thus say that “the derivative of sine is cosine,” and “the derivative of cosine is minus sine.” Notice that the second derivatives satisfy ′′ ′′ (sin x) =− sin x, (cos x) =− cos x. 0.5.3 Exponential and natural logarithm functions x The derivative of e and ln x are 1 x ′ x ′ (e ) = e , (ln x) = . x 0.6 Definition of the integral The definite integral of a function f(x) 0 from x = a to b (b a) is defined as the area bounded by the vertical lines x = a, x = b, the x-axis and the curve y = f(x). This “area under the curve” is obtained by a limit. First, the area is approximated by a sum of rectangle areas. Second, the integral is defined to be the limit of the rectangle areas as the width of each individual rectangle goes to zero and the number of rectangles goes to infinity. This resulting infinite sum is called a Riemann Sum, and we define Z N b f(x)dx = lim f(a+(n− 1)h)· h, (2) å a h→0 n=1 where N = (b− a)/h is the number of terms in the sum. The symbols on the left- hand-side of (2) are read as “the integral from a to b of f of x dee x.” The Riemann Sum definition is extended to all values of a and b and for all values of f(x) (positive and negative). Accordingly, Z Z Z Z a b b b f(x)dx =− f(x)dx and (− f(x))dx =− f(x)dx. b a a a Also, Z Z Z c b c f(x)dx = f(x)dx+ f(x)dx, a a b which states when f(x) 0 and a b c that the total area is equal to the sum of its parts. CHAPTER 0. A SHORT MATHEMATICAL REVIEW 30.7. THE FUNDAMENTAL THEOREM OF CALCULUS 0.7 The fundamental theorem of calculus View tutorial on YouTube Using the definition of the derivative, we differentiate the following integral: R R Z x+h x x f(s)ds− f(s)ds d a a f(s)ds= lim dx h→0 h a R x+h f(s)ds x = lim h h→0 h f(x) = lim h h→0 = f(x). This result is called the fundamental theorem of calculus, and provides a connection between differentiation and integration. The fundamental theorem teaches us how to integrate functions. Let F(x) be a ′ function such that F (x)= f(x). We say that F(x) is an antiderivative of f(x). Then from the fundamental theorem and the fact that the derivative of a constant equals zero, Z x F(x)= f(s)ds+ c. a R b Now, F(a) = c and F(b) = f(s)ds+ F(a). Therefore, the fundamental theorem a shows us how to integrate a function f(x) provided we can find its antiderivative: Z b f(s)ds= F(b)− F(a). (3) a Unfortunately, finding antiderivatives is much harder than finding derivatives, and indeed, most complicated functions cannot be integrated analytically. We can also derive the very important result (3) directly from the definition of the derivative (1) and the definite integral (2). We will see it is convenient to choose ′ the same h in both limits. With F (x)= f(x), we have Z Z b b ′ f(s)ds= F (s)ds a a N ′ = lim F (a+(n− 1)h)· h å h→0 n=1 N F(a+ nh)− F(a+(n− 1)h) = lim · h å h h→0 n=1 N = lim F(a+ nh)− F(a+(n− 1)h). å h→0 n=1 The last expression has an interesting structure. All the values of F(x) evaluated at the points lying between the endpoints a and b cancel each other in consecutive terms. Only the value− F(a) survives when n = 1, and the value +F(b) when n= N, yielding again (3). 4 CHAPTER 0. A SHORT MATHEMATICAL REVIEW0.8. DEFINITE AND INDEFINITE INTEGRALS 0.8 Definite and indefinite integrals The Riemann sum definition of an integral is called a definite integral. It is convenient to also define an indefinite integral by Z f(x)dx = F(x), where F(x) is the antiderivative of f(x). 0.9 Indefinite integrals of elementary functions From our known derivatives of elementary functions, we can determine some sim- ple indefinite integrals. The power rule gives us Z n+1 x n x dx = + c, n̸=− 1. n+ 1 When n=− 1, and x is positive, we have Z 1 dx = ln x+ c. x If x is negative, using the chain rule we have d 1 ln(− x)= . dx x Therefore, since  − x if x 0; x= x if x 0, we can generalize our indefinite integral to strictly positive or strictly negative x: Z 1 dx = lnx+ c. x Trigonometric functions can also be integrated: Z Z cos xdx = sin x+ c, sin xdx =− cos x+ c. Easily proved identities are an addition rule: Z Z Z f(x)+ g(x) dx = f(x)dx+ g(x)dx; ( ) and multiplication by a constant: Z Z A f(x)dx = A f(x)dx. This permits integration of functions such as Z 3 2 x 7x 2 (x + 7x+ 2)dx = + + 2x+ c, 3 2 and Z (5 cos x+ sin x)dx = 5 sin x− cos x+ c. CHAPTER 0. A SHORT MATHEMATICAL REVIEW 50.10. SUBSTITUTION 0.10 Substitution More complicated functions can be integrated using the chain rule. Since d ′ ′ f(g(x))= f (g(x))· g (x), dx we have Z ′ ′ f g(x) · g (x)dx = f g(x) + c. ( ) ( ) This integration formula is usually implemented by letting y = g(x). Then one ′ writes dy= g (x)dx to obtain Z Z ′ ′ ′ f g(x) g (x)dx = f (y)dy ( ) = f(y)+ c = f(g(x))+ c. 0.11 Integration by parts Another integration technique makes use of the product rule for differentiation. Since ′ ′ ′ ( f g) = f g+ f g , we have ′ ′ ′ f g=( f g)− f g . Therefore, Z Z ′ ′ f (x)g(x)dx = f(x)g(x)− f(x)g (x)dx. Commonly, the above integral is done by writing ′ u= g(x) dv= f (x)dx ′ du= g (x)dx v= f(x). Then, the formula to be memorized is Z Z udv= uv− vdu. 0.12 Taylor series A Taylor series of a function f(x) about a point x = a is a power series repre- sentation of f(x) developed so that all the derivatives of f(x) at a match all the derivatives of the power series. Without worrying about convergence here, we have ′′ ′′′ f (a) f (a) ′ 2 3 f(x)= f(a)+ f (a)(x− a)+ (x− a) + (x− a) + . . . . 2 3 Notice that the first term in the power series matches f(a), all other terms vanishing, ′ the second term matches f (a), all other terms vanishing, etc. Commonly, the Taylor 6 CHAPTER 0. A SHORT MATHEMATICAL REVIEW0.13. FUNCTIONS OF SEVERAL VARIABLES series is developed with a = 0. We will also make use of the Taylor series in a slightly different form, with x = x +e and a= x : ′′ ′′′ f (x ) f (x ) ′ 2 3 f(x +e)= f(x )+ f (x )e+ e + e + . . . . 2 3 Another way to view this series is that of g(e)= f(x +e), expanded about e= 0. Taylor series that are commonly used include 2 3 x x x e = 1+ x+ + + . . . , 2 3 3 5 x x sin x = x− + − . . . , 3 5 2 4 x x cos x = 1− + − . . . , 2 4 1 2 = 1− x+ x − . . . , forx 1, 1+ x 2 3 x x ln(1+ x)= x− + − . . . , forx 1. 2 3 0.13 Functions of several variables For simplicity, we consider a function f = f(x, y) of two variables, though the results are easily generalized. The partial derivative of f with respect to x is defined as ¶ f f(x+ h, y)− f(x, y) = lim , ¶x h h→0 and similarly for the partial derivative of f with respect to y. To take the partial derivative of f with respect to x, say, take the derivative of f with respect to x holding y fixed. As an example, consider 3 2 3 f(x, y)= 2x y + y . We have ¶ f ¶ f 2 2 3 2 = 6x y , = 4x y+ 3y . ¶x ¶y Second derivatives are defined as the derivatives of the first derivatives, so we have 2 2 ¶ f ¶ f 2 3 = 12xy , = 4x + 6y; 2 2 ¶x ¶y and the mixed second partial derivatives are 2 2 ¶ f ¶ f 2 2 = 12x y, = 12x y. ¶x¶y ¶y¶x In general, mixed partial derivatives are independent of the order in which the derivatives are taken. Partial derivatives are necessary for applying the chain rule. Consider d f = f(x+ dx, y+ dy)− f(x, y). CHAPTER 0. A SHORT MATHEMATICAL REVIEW 70.14. COMPLEX NUMBERS We can write d f as d f = f(x+ dx, y+ dy)− f(x, y+ dy)+ f(x, y+ dy)− f(x, y) ¶ f ¶ f = dx+ dy. ¶x ¶y If one has f = f(x(t), y(t)), say, then d f ¶ f dx ¶ f dy = + . dt ¶x dt ¶y dt And if one has f = f(x(r,q), y(r,q)), say, then ¶ f ¶ f ¶x ¶ f ¶y ¶ f ¶ f ¶x ¶ f ¶y = + , = + . ¶r ¶x ¶r ¶y ¶r ¶q ¶x ¶q ¶y ¶q A Taylor series of a function of several variables can also be developed. Here, all partial derivatives of f(x, y) at (a, b) match all the partial derivatives of the power series. With the notation 2 2 2 ¶ f ¶ f ¶ f ¶ f ¶ f f = , f = , f = , f = , f = , etc., x y xx xy yy 2 2 ¶x ¶y ¶x ¶x¶y ¶y we have f(x, y)= f(a, b)+ f (a, b)(x− a)+ f (a, b)(y− b) x y   1 2 2 + f (a, b)(x− a) + 2 f (a, b)(x− a)(y− b)+ f (a, b)(y− b) + . . . xx xy yy 2 0.14 Complex numbers View tutorial on YouTube: Complex Numbers View tutorial on YouTube: Complex Exponential Function We define the imaginary number i to be one of the two numbers that satisfies the √ 2 rule(i) =− 1, the other number being− i. Formally, we write i = − 1. A complex number z is written as z= x+ iy, where x and y are real numbers. We call x the real part of z and y the imaginary part and write x = Re z, y= Im z. Two complex numbers are equal if and only if their real and imaginary parts are equal. ¯ The complex conjugate of z= x+ iy, denoted as z, is defined as z ¯ = x− iy. Using z and z ¯, we have 1 1 Re z= (z+ z ¯) , Im z= (z− z ¯) . (4) 2 2i 8 CHAPTER 0. A SHORT MATHEMATICAL REVIEW0.14. COMPLEX NUMBERS Furthermore, ¯ zz=(x+ iy)(x− iy) 2 2 2 = x − i y 2 2 = x + y ; and we define the absolute value of z, also called the modulus of z, by 1/2 z=(zz ¯) q 2 2 = x + y . We can add, subtract, multiply and divide complex numbers to get new complex numbers. With z= x+ iy and w= s+ it, and x, y, s, t real numbers, we have z+ w=(x+ s)+ i(y+ t); z− w=(x− s)+ i(y− t); zw=(x+ iy)(s+ it) =(xs− yt)+ i(xt+ ys); z zw ¯ = ¯ w ww (x+ iy)(s− it) = 2 2 s + t (xs+ yt) (ys− xt) = + i . 2 2 2 2 s + t s + t Furthermore, q 2 2 zw= (xs− yt) +(xt+ ys) q 2 2 2 2 = (x + y )(s + t ) =zw; and zw=(xs− yt)− i(xt+ ys) =(x− iy)(s− it) = z ¯w ¯ . Similarly z z z z ¯ = , ( )= . w w w w ¯ Also, z+ w= z+ w. However,z+ w≤ z+w, a theorem known as the triangle inequality. It is especially interesting and useful to consider the exponential function of an imaginary argument. Using the Taylor series expansion of an exponential function, we have 2 3 4 5 (iq) (iq) (iq) (iq) iq e = 1+(iq)+ + + + . . . 2 3 4 5     2 4 3 5 q q q q = 1− + − . . . + i q− + + . . . 2 4 3 5 = cosq+ i sinq. CHAPTER 0. A SHORT MATHEMATICAL REVIEW 90.14. COMPLEX NUMBERS Since we have determined that iq iq cosq = Re e , sinq = Im e , (5) we also have using (4) and (5), the frequently used expressions iq − iq iq − iq e + e e − e cosq = , sinq = . 2 2i iq The much celebrated Euler’s identity derives from e = cosq+ i sinq by setting q = p, and using cosp =− 1 and sinp = 0: ip e + 1= 0, and this identity links the five fundamental numbers, 0, 1, i, e and p, using three basic mathematical operations, addition, multiplication and exponentiation, only once. The complex number z can be represented in the complex plane with Re z as the x-axis and Im z as the y-axis. This leads to the polar representation of z= x+ iy: iq z= re , where r =z and tanq = y/x. We define arg z = q. Note that q is not unique, though it is conventional to choose the value such that− p q≤ p, and q = 0 when r = 0. iq Useful trigonometric relations can be derived using e and properties of the exponential function. The addition law can be derived from i(x+y) ix iy e = e e . We have cos(x+ y)+ i sin(x+ y)=(cos x+ i sin x)(cos y+ i sin y) =(cos x cos y− sin x sin y)+ i(sin x cos y+ cos x sin y); yielding cos(x+ y)= cos x cos y− sin x sin y, sin(x+ y)= sin x cos y+ cos x sin y. inq iq n De Moivre’s Theorem derives from e =(e ) , yielding the identity n cos(nq)+ i sin(nq)=(cosq+ i sinq) . For example, if n= 2, we derive 2 cos 2q+ i sin 2q =(cosq+ i sinq) 2 2 =(cos q− sin q)+ 2i cosq sinq. Therefore, 2 2 cos 2q = cos q− sin q, sin 2q = 2 cosq sinq. 2 2 With a little more manipulation using cos q+ sin q = 1, we can derive 1+ cos 2q 1− cos 2q 2 2 cos q = , sin q = , 2 2 10 CHAPTER 0. A SHORT MATHEMATICAL REVIEW0.14. COMPLEX NUMBERS which are useful formulas for determining Z Z 1 1 2 2 cos q dq = (2q+ sin 2q)+ c, sin q dq = (2q− sin 2q)+ c, 4 4 from which follows Z Z 2p 2p 2 2 sin q dq = cos q dq = p. 0 0 CHAPTER 0. A SHORT MATHEMATICAL REVIEW 110.14. COMPLEX NUMBERS 12 CHAPTER 0. A SHORT MATHEMATICAL REVIEW