Introduction to methods of applied mathe

Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers Sean Mauch http://www.its.caltech.edu/˜sean January 24, 2004 Contents Anti-Copyright xxiv Preface 0.1 Advice to Teachers 0.2 Acknowledgments 0.3 Warnings and Disclaimers 0.4 Suggested Use 0.5 About the Title xxv xxv xxv xxvi xxvii xxvii I Algebra Sets 1.1 1.2 1.3 1.4 1.5 1.6 1.7 and Functions Sets Single Valued Functions Inverses and Multi-Valued Functions Transforming Equations Exercises Hints Solutions i 2 11 14 16 Vectors 2.1 Vectors 2.1.1 Scalars and Vectors 2.1.2 The Kronecker Delta and Einstein 2.1.3 The Dot and Cross Product 2.2 Sets of Vectors in n Dimensions 2.3 Exercises 2.4 Hints 2.5 Solutions II Summation Convention Calculus Differential Calculus 3.1 Limits of Functions 3.2 Continuous Functions 3.3 The Derivative 3.4 Implicit Differentiation 3.5 Maxima and Minima 3.6 Mean Value Theorems 3.6.1 Application: Using Taylor’s Theorem to 3.6.2 Application: Finite Difference Schemes 3.7 L’Hospital’s Rule 3.8 Exercises 3.8.1 Limits of Functions 3.8.2 Continuous Functions 3.8.3 The Derivative 3.8.4 Implicit Differentiation 3.8.5 Maxima and Minima 3.8.6 Mean Value Theorems 22 22 22 25 26 33 36 38 40 47 Approximate ii Functions 48 48 53 56 61 62 66 68 73 75 81 81 81 82 84 84 85 3.8.7 L’Hospital’s Rule 3.9 Hints 3.10 Solutions 3.11 Quiz 3.12 Quiz Solutions 85 87 93 113 114 116 116 122 122 123 125 127 127 130 134 134 134 136 136 137 138 141 150 151 Vector Calculus 5.1 Vector Functions 5.2 Gradient, Divergence and Curl 5.3 Exercises 154 154 155 163 Integral Calculus 4.1 The Indefinite Integral 4.2 The Definite Integral 4.2.1 Definition 4.2.2 Properties 4.3 The Fundamental Theorem of Integral Calculus 4.4 Techniques of Integration 4.4.1 Partial Fractions 4.5 Improper Integrals 4.6 Exercises 4.6.1 The Indefinite Integral 4.6.2 The Definite Integral 4.6.3 The Fundamental Theorem of Integration 4.6.4 Techniques of Integration 4.6.5 Improper Integrals 4.7 Hints 4.8 Solutions 4.9 Quiz 4.10 Quiz Solutions iii 5.4 Hints 5.5 Solutions 5.6 Quiz 5.7 Quiz Solutions III Functions of a Complex Variable Complex Numbers 6.1 Complex Numbers 6.2 The Complex Plane 6.3 Polar Form 6.4 Arithmetic and Vectors 6.5 Integer Exponents 6.6 Rational Exponents 6.7 Exercises 6.8 Hints 6.9 Solutions 166 168 177 178 179 Functions of a Complex Variable 7.1 Curves and Regions 7.2 The Point at Infinity and the Stereographic 7.3 A Gentle Introduction to Branch Points 7.4 Cartesian and Modulus-Argument Form 7.5 Graphing Functions of a Complex Variable 7.6 Trigonometric Functions 7.7 Inverse Trigonometric Functions 7.8 Riemann Surfaces 7.9 Branch Points 7.10 Exercises 180 180 184 188 193 195 197 201 208 211 Projection 239 239 242 246 246 249 252 259 268 270 286 iv 7.11 Hints 297 7.12 Solutions 302 Analytic Functions 8.1 Complex Derivatives 8.2 Cauchy-Riemann Equations 8.3 Harmonic Functions 8.4 Singularities 8.4.1 Categorization of Singularities 8.4.2 Isolated and Non-Isolated Singularities 8.5 Application: Potential Flow 8.6 Exercises 8.7 Hints 8.8 Solutions 360 360 367 372 377 377 381 383 388 396 399 Analytic Continuation 9.1 Analytic Continuation 9.2 Analytic Continuation of Sums 9.3 Analytic Functions Defined in Terms of Real Variables 9.3.1 Polar Coordinates 9.3.2 Analytic Functions Defined in Terms of Their Real or Imaginary Parts 9.4 Exercises 9.5 Hints 9.6 Solutions 437 437 440 442 446 450 454 456 457 462 462 464 466 467 10 Contour Integration and the Cauchy-Goursat Theorem 10.1 Line Integrals 10.2 Contour Integrals 10.2.1 Maximum Modulus Integral Bound 10.3 The Cauchy-Goursat Theorem v 10.4 10.5 10.6 10.7 Contour Deformation Morera’s Theorem Indefinite Integrals Fundamental Theorem of Calculus via 10.7.1 Line Integrals and Primitives 10.7.2 Contour Integrals 10.8 Fundamental Theorem of Calculus via 10.9 Exercises 10.10Hints 10.11Solutions 11 Cauchy’s Integral Formula 11.1 Cauchy’s Integral Formula 11.2 The Argument Theorem 11.3 Rouche’s Theorem 11.4 Exercises 11.5 Hints 11.6 Solutions Primitives Complex Calculus 12 Series and Convergence 12.1 Series of Constants 12.1.1 Definitions 12.1.2 Special Series 12.1.3 Convergence Tests 12.2 Uniform Convergence 12.2.1 Tests for Uniform Convergence 12.2.2 Uniform Convergence and Continuous Functions 12.3 Uniformly Convergent Power Series 12.4 Integration and Differentiation of Power Series 12.5 Taylor Series vi 469 471 473 474 474 474 475 478 482 483 493 494 501 502 505 509 511 525 525 525 527 529 536 537 539 539 547 550 12.6 12.7 12.8 12.9 12.5.1 Newton’s Binomial Formula Laurent Series Exercises 12.7.1 Series of Constants 12.7.2 Uniform Convergence 12.7.3 Uniformly Convergent Power Series 12.7.4 Integration and Differentiation of Power Series 12.7.5 Taylor Series 12.7.6 Laurent Series Hints Solutions 13 The Residue Theorem 13.1 The Residue Theorem 13.2 Cauchy Principal Value for Real Integrals 13.2.1 The Cauchy Principal Value 13.3 Cauchy Principal Value for Contour Integrals 13.4 Integrals on the Real Axis 13.5 Fourier Integrals 13.6 Fourier Cosine and Sine Integrals 13.7 Contour Integration and Branch Cuts 13.8 Exploiting Symmetry 13.8.1 Wedge Contours 13.8.2 Box Contours 13.9 Definite Integrals Involving Sine and Cosine 13.10Infinite Sums 13.11Exercises 13.12Hints 13.13Solutions vii 553 555 560 560 566 566 568 569 571 574 582 626 626 634 634 639 643 647 649 652 655 655 658 659 662 666 680 686 IV Ordinary Differential Equations 772 14 First Order Differential Equations 14.1 Notation 14.2 Example Problems 14.2.1 Growth and Decay 14.3 One Parameter Families of Functions 14.4 Integrable Forms 14.4.1 Separable Equations 14.4.2 Exact Equations 14.4.3 Homogeneous Coefficient Equations 14.5 The First Order, Linear Differential Equation 14.5.1 Homogeneous Equations 14.5.2 Inhomogeneous Equations 14.5.3 Variation of Parameters 14.6 Initial Conditions 14.6.1 Piecewise Continuous Coefficients and Inhomogeneities 14.7 Well-Posed Problems 14.8 Equations in the Complex Plane 14.8.1 Ordinary Points 14.8.2 Regular Singular Points 14.8.3 Irregular Singular Points 14.8.4 The Point at Infinity 14.9 Additional Exercises 14.10Hints 14.11Solutions 14.12Quiz 14.13Quiz Solutions viii 773 773 775 775 777 779 780 782 786 791 791 792 795 796 797 801 803 803 806 812 814 816 819 822 843 844 15 First Order Linear Systems of Differential Equations 15.1 Introduction 15.2 Using Eigenvalues and Eigenvectors to find Homogeneous Solutions 15.3 Matrices and Jordan Canonical Form 15.4 Using the Matrix Exponential 15.5 Exercises 15.6 Hints 15.7 Solutions 846 846 847 852 860 865 870 872 16 Theory of Linear Ordinary Differential Equations 16.1 Exact Equations 16.2 Nature of Solutions 16.3 Transformation to a First Order System 16.4 The Wronskian 16.4.1 Derivative of a Determinant 16.4.2 The Wronskian of a Set of Functions 16.4.3 The Wronskian of the Solutions to a Differential Equation 16.5 Well-Posed Problems 16.6 The Fundamental Set of Solutions 16.7 Adjoint Equations 16.8 Additional Exercises 16.9 Hints 16.10Solutions 16.11Quiz 16.12Quiz Solutions 900 900 901 905 905 905 906 908 911 913 915 919 920 922 928 929 17 Techniques for Linear Differential 17.1 Constant Coefficient Equations 17.1.1 Second Order Equations 17.1.2 Real-Valued Solutions Equations 930 930 931 935 ix particularly games of chance Never explain mathematics to the layperson T.2 Playing the Odds Years ago in a classroom not so far away, your author was being subjected to a presentation of a lengthy proof About five minutes into the lecture, the entire class was hopelessly lost At the forty-five minute mark the professor had a combinatorial expression that covered most of a chalk board From his previous queries the professor knew that none of the students had a clue what was going on This pleased him and he had became more animated as the lecture had progressed He gestured to the board with a smirk and asked for the value of the expression Without a moment’s hesitation, I nonchalantly replied, “zero” The professor was taken aback He was clearly impressed that I was able to evaluate the expression, especially because I had done it in my head and so quickly He enquired as to my method “Probability”, I replied “Professors often present difficult problems that have simple, elegant solutions Zero is the most elegant of numerical answers and thus most likely to be the correct answer My second guess would have been one.” The professor was not amused Whenever a professor asks the class a question which has a numeric answer, immediately respond, “zero” If you are asked about your method, casually say something vague about symmetry Speak with confidence and give non-verbal cues that you consider the problem to be elementary This tactic will usually suffice It’s quite likely that some kind of symmetry is involved And if it isn’t your response will puzzle the professor They may continue with the next topic, not wanting to admit that they don’t see the “symmetry” in such an elementary problem If they press further, start mumbling to yourself Pretend that you are lost in thought, perhaps considering some generalization of the result They may be a little irked that you are ignoring them, but it’s better than divulging your true method 2279 Appendix U Economics There are two important concepts in economics The first is “Buy low, sell high”, which is self-explanitory The second is opportunity cost, the highest valued alternative that must be sacrificed to attain something or otherwise satisfy a want I discovered this concept as an undergraduate at Caltech I was never very interested in computer games, but one day I found myself randomly playing tetris Out of the blue I was struck by a revelation: “I could be having sex right now.” I haven’t played a computer game since 2280 Appendix V Glossary Phrases often have different meanings in mathematics than in everyday usage Here I have collected definitions of some mathematical terms which might confuse the novice beyond the scope of this text: Beyond the comprehension of the author difficult: Essentially impossible Note that mathematicians never refer to problems they have solved as being difficult This would either be boastful, (claiming that you can solve difficult problems), or self-deprecating, (admitting that you found the problem to be difficult) interesting: This word is grossly overused in math and science It is often used to describe any work that the author has done, regardless of the work’s significance or novelty It may also be used as a synonym for difficult It has a completely different meaning when used by the non-mathematician When I tell people that I am a mathematician they typically respond with, “That must be interesting.”, which means, “I don’t know anything about math or what mathematicians do.” I typically answer, “No Not really.” non-obvious or non-trivial: Real fuckin’ hard one can prove that : The “one” that proved it was a genius like Gauss The phrase literally means “you haven’t got a chance in hell of proving that ” 2281 simple: Mathematicians communicate their prowess to colleagues and students by referring to all problems as simple or trivial If you ever become a math professor, introduce every example as being “really quite trivial.” Here are some less interesting words and phrases that you are probably already familiar with corollary: a proposition inferred immediately from a proved proposition with little or no additional proof lemma: an auxiliary proposition used in the demonstration of another proposition theorem: a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions For even more fun say it in your best Elmer Fudd accent “This next pwobwem is weawy quite twiviaw” 2282 Appendix W whoami 2283 Figure W.1: Graduation, June 13, 2003 2284 Index a + i b form, 184 Abel’s formula, 910 absolute convergence, 526 adjoint of a differential operator, 915 of operators, 1314 analytic, 361 Analytic continuation Fourier integrals, 1550 analytic continuation, 437 analytic functions, 2225 anti-derivative, 473 Argand diagram, 184 argument of a complex number, 186 argument theorem, 501 asymptotic expansions, 1251 integration by parts, 1263 asymptotic relations, 1251 autonomous D.E., 992 average value theorem, 499 Bessel functions, 1622 generating function, 1629 of the first kind, 1628 second kind, 1644 Bessel’s equation, 1622 Bessel’s Inequality, 1296 Bessel’s inequality, 1340 bilinear concomitant, 917 binomial coefficients, 2276 binomial formula, 2276 boundary value problems, 1109 branch principal, branch point, 270 branches, calculus of variations, 2060 canonical forms constant coefficient equation, 1018 of differential equations, 1018 cardinality of a set, Cartesian form, 184 Bernoulli equations, 984 2285 Cartesian product of sets, Cauchy convergence, 526 Cauchy principal value, 634, 1548 Cauchy’s inequality, 497 Cauchy-Riemann equations, 367, 2225 chicken spherical, 2277 clockwise, 241 closed interval, closure relation and Fourier transform, 1552 discrete sets of functions, 1297 codomain, comparison test, 529 completeness of sets of functions, 1297 sets of vectors, 34 complex conjugate, 182, 184 complex derivative, 360, 361 complex infinity, 242 complex number, 182 magnitude, 185 modulus, 185 complex numbers, 180 arithmetic, 193 set of, vectors, 193 complex plane, 184 first order differential equations, 803 computer games, 2280 connected region, 240 constant coefficient differential equations, 930 continuity, 53 uniform, 55 continuous piecewise, 55 continuous functions, 53, 536, 539 contour, 240 traversal of, 241 contour integral, 465 convergence absolute, 526 Cauchy, 526 comparison test, 529 Gauss’ test, 536 in the mean, 1296 integral test, 530 of integrals, 1470 Raabe’s test, 535 ratio test, 531 root test, 533 sequences, 525 series, 526 uniform, 536 convolution theorem and Fourier transform, 1554 for Laplace transforms, 1490 convolutions, 1490 counter-clockwise, 241 2286 curve, 240 closed, 240 continuous, 240 Jordan, 240 piecewise smooth, 240 simple, 240 smooth, 240 equidimensional-in-x, 995 equidimensional-in-y, 997 Euler, 940 exact, 782, 945 first order, 773, 791 homogeneous, 774 homogeneous coefficient, 786 inhomogeneous, 774 linear, 774 order, 773 ordinary, 773 scale-invariant, 1000 separable, 780 without explicit dep on y, 946 differential operator linear, 902 Dirac delta function, 1041, 1298 direction negative, 241 positive, 241 directional derivative, 157 discontinuous functions, 54, 1337 discrete derivative, 1167 discrete integral, 1167 disjoint sets, domain, definite integral, 122 degree of a differential equation, 774 del, 157 delta function Kronecker, 34 derivative complex, 361 determinant derivative of, 905 difference of sets, difference equations constant coefficient equations, 1174 exact equations, 1168 first order homogeneous, 1169 first order inhomogeneous, 1171 differential calculus, 48 differential equations autonomous, 992 constant coefficient, 930 degree, 774 economics, 2280 eigenfunctions, 1330 eigenvalue problems, 1330 2287 eigenvalues, 1330 elements of a set, empty set, entire, 361 equidimensional differential equations, 940 equidimensional-in-x D.E., 995 equidimensional-in-y D.E., 997 Euler differential equations, 940 Euler’s formula, 189 Euler’s notation i, 182 Euler’s theorem, 786 Euler-Mascheroni constant, 1611 exact differential equations, 945 exact equations, 782 exchanging dep and indep var., 990 extended complex plane, 242 extremum modulus theorem, 500 table of, 2254 Fourier series, 1330 and Fourier transform, 1539 uniform convergence, 1353 Fourier Sine series, 1345 Fourier sine series, 1429 Fourier sine transform, 1563 of derivatives, 1564 table of, 2255 Fourier transform alternate definitions, 1544 closure relation, 1552 convolution theorem, 1554 of a derivative, 1553 Parseval’s theorem, 1557 shift property, 1559 table of, 2250, 2253 Fredholm alternative theorem, 1109 Fredholm equations, 1027 Frobenius series first order differential equation, 808 function bijective, injective, inverse of, multi-valued, single-valued, surjective, function elements, 437 functional equation, 389 Fibonacci sequence, 1179 fluid flow ideal, 383 formally self-adjoint operators, 1315 Fourier coefficients, 1291, 1335 behavior of, 1349 Fourier convolution theorem, 1554 Fourier cosine series, 1344 Fourier cosine transform, 1562 of derivatives, 1564 2288 fundamental set of solutions of a differential equation, 913 fundamental theorem of algebra, 498 fundamental theorem of calculus, 125 homogeneous solutions of differential equations, 902 i Euler’s notation, 182 ideal fluid flow, 383 identity map, ill-posed problems, 801 linear differential equations, 911 image of a mapping, imaginary number, 182 imaginary part, 182 improper integrals, 130 indefinite integral, 116, 473 indicial equation, 1201 infinity complex, 242 first order differential equation, 814 point at, 242 inhomogeneous differential equations, 774 initial conditions, 796 inner product of functions, 1288 integers set of, integral bound maximum modulus, 467 integral calculus, 116 integral equations, 1027 gambler’s ruin problem, 1166, 1175 Gamma function, 1605 difference equation, 1605 Euler’s formula, 1605 Gauss’ formula, 1609 Hankel’s formula, 1607 Weierstrass’ formula, 1611 Gauss’ test, 536 generating function for Bessel functions, 1628 geometric series, 527 Gibb’s phenomenon, 1358 gradient, 157 Gramm-Schmidt orthogonalization, 1284 greatest integer function, Green’s formula, 917, 1315 harmonic conjugate, 372 harmonic series, 528, 564 Heaviside function, 797, 1041 holomorphic, 361 homogeneous coefficient equations, 786 homogeneous differential equations, 774 homogeneous functions, 786 homogeneous solution, 793 2289 boundary value problems, 1027 initial value problems, 1027 integrals improper, 130 integrating factor, 792 integration techniques of, 127 intermediate value theorem, 55 intersection of sets, interval closed, open, inverse function, inverse image, irregular singular points, 1216 first order differential equations, 812 convolution theorem, 1490 of derivatives, 1490 Laurent expansions, 627, 2225 Laurent series, 555, 2227 first order differential equation, 808 leading order behavior for differential equations, 1255 least integer function, least squares fit Fourier series, 1337 Legendre polynomials, 1285 limit left and right, 50 limits of functions, 48 line integral, 463 complex, 465 linear differential equations, 774 linear differential operator, 902 linear space, 1278 Liouville’s theorem, 497 j electrical engineering, 182 Jordan curve, 240 Jordan’s lemma, 2226 magnitude, 185 maximum modulus integral bound, 467 maximum modulus theorem, 500 Mellin inversion formula, 1478 minimum modulus theorem, 500 modulus, 185 multi-valued function, Kramer’s rule, 2270 Kronecker delta function, 34 L’Hospital’s rule, 75 Lagrange’s identity, 917, 949, 1314 Laplace transform inverse, 1477 Laplace transform pairs, 1479 Laplace transforms, 1475 nabla, 157 2290 natural boundary, 437 Newton’s binomial formula, 2276 norm of functions, 1288 normal form of differential equations, 1021 null vector, 24 of differential equations, 903 periodic extension, 1336 piecewise continuous, 55 point at infinity, 242 differential equations, 1216 polar form, 188 potential flow, 383 power series definition of, 539 differentiation of, 547 integration of, 547 radius of convergence, 541 uniformly convergent, 539 principal argument, 186 principal branch, principal root, 199 principal value, 634, 1548 pure imaginary number, 182 one-to-one mapping, open interval, opportunity cost, 2280 optimal asymptotic approximations, 1268 order of a differential equation, 773 of a set, ordinary points first order differential equations, 803 of linear differential equations, 1184 orthogonal series, 1291 orthogonality weighting functions, 1290 orthonormal, 1288 Raabe’s test, 535 range of a mapping, ratio test, 531 rational numbers set of, Rayleigh’s quotient, 1426 minimum property, 1426 real numbers set of, real part, 182 Parseval’s equality, 1340 Parseval’s theorem for Fourier transform, 1557 partial derivative, 155 particular solution, 793 of an ODE, 1059 particular solutions 2291 rectangular unit vectors, 24 reduction of order, 947 and the adjoint equation, 948 difference equations, 1177 region connected, 240 multiply-connected, 240 simply-connected, 240 regular, 361 regular singular points first order differential equations, 806 regular Sturm-Liouville problems, 1420 properties of, 1428 residuals of series, 527 residue theorem, 631, 2226 principal values, 643 residues, 627, 2225 of a pole of order n, 627, 2226 Riccati equations, 986 Riemann zeta function, 528 Riemann-Lebesgue lemma, 1471 root test, 533 Rouche’s theorem, 502 series, 525 comparison test, 529 convergence of, 525, 526 Gauss’ test, 536 geometric, 527 integral test, 530 Raabe’s test, 535 ratio test, 531 residuals, 527 root test, 533 tail of, 526 set, similarity transformation, 1888 single-valued function, singularity, 377 branch point, 377 spherical chicken, 2277 stereographic projection, 243 Stirling’s approximation, 1613 subset, proper, Taylor series, 550, 2226 first order differential equations, 805 table of, 2241 transformations of differential equations, 1018 of independent variable, 1024 to constant coefficient equation, 1025 to integral equations, 1027 scalar field, 155 scale-invariant D.E., 1000 separable equations, 780 sequences convergence of, 525 2292 trigonometric identities, 2264 uniform continuity, 55 uniform convergence, 536 of Fourier series, 1353 of integrals, 1470 union of sets, variation of parameters first order equation, 795 vector components of, 25 rectangular unit, 24 vector calculus, 154 vector field, 155 vector-valued functions, 154 Volterra equations, 1027 wave equation D’Alembert’s solution, 1933 Fourier transform solution, 1933 Laplace transform solution, 1934 Weber’s function, 1644 Weierstrass M-test, 537 well-posed problems, 801 linear differential equations, 911 Wronskian, 906, 907 zero vector, 24 2293