Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 8 docx

I.5 Table of Fourier Cosine Transforms f(x) 1 π  ∞ 0 f(x) cos (ωx) dx 2  ∞ 0 C(ω) cos (ωx) dω C(ω) f  (x) ωS(ω) − 1 π f(0) f  (x) −ω 2 C(ω) − 1 π f  (0) xf(x) ∂ ∂ω F s [f(x)] f(cx), c > 0 1 c C  ω c  2c x 2 + c 2 e −cω e −cx c/π ω 2 + c 2 e −cx 2 1 √ 4πc e −ω 2 /(4c)  π c e −x 2 /(4c) e −cω 2 2254 I.6 Table of Fourier Sine Transforms f(x) 1 π  ∞ 0 f(x) sin (ωx) dx 2  ∞ 0 S(ω) sin (ωx) dω S(ω) f  (x) −ωC(ω) f  (x) −ω 2 S(ω) + 1 π ωf(0) xf(x) − ∂ ∂ω F c [f(x)] f(cx), c > 0 1 c S  ω c  2x x 2 + c 2 e −cω e −cx ω/π ω 2 + c 2 2 arctan  x c  1 ω e −cω 1 x e −cx 1 π arctan  ω c  2255 1 1 πω 2 x 1 x e −cx 2 ω 4c 3/2 √ π e −ω 2 /(4c) √ πx 2c 3/2 e −x 2 /(4c) ω e −cω 2 2256 Appendix J Table of Wronskians W [x − a, x − b] b − a W  e ax , e bx  (b − a) e (a+b)x W [cos(ax), sin(ax)] a W [cosh(ax), sinh(ax)] a W [ e ax cos(bx), e ax sin(bx)] b e 2ax W [ e ax cosh(bx), e ax sinh(bx)] b e 2ax W [sin(c(x − a)), sin(c(x − b))] c sin(c(b − a)) W [cos(c(x − a)), cos(c(x − b))] c sin(c(b − a)) W [sin(c(x − a)), cos(c(x − b))] −c cos(c(b − a)) 2257 W [sinh(c(x − a)), sinh(c(x − b))] c sinh(c(b − a)) W [cosh(c(x − a)), cosh(c(x − b))] c cosh(c(b − a)) W [sinh(c(x − a)), cosh(c(x − b))] −c cosh(c(b − a)) W  e dx sin(c(x − a)), e dx sin(c(x − b))  c e 2dx sin(c(b − a)) W  e dx cos(c(x − a)), e dx cos(c(x − b))  c e 2dx sin(c(b − a)) W  e dx sin(c(x − a)), e dx cos(c(x − b))  −c e 2dx cos(c(b − a)) W  e dx sinh(c(x − a)), e dx sinh(c(x − b))  c e 2dx sinh(c(b − a)) W  e dx cosh(c(x − a)), e dx cosh(c(x − b))  −c e 2dx sinh(c(b − a)) W  e dx sinh(c(x − a)), e dx cosh(c(x − b))  −c e 2dx cosh(c(b − a)) W [(x − a) e cx , (x − b) e cx ] (b − a) e 2cx 2258 Appendix K Sturm-Liouville Eigenvalue Problems • y  + λ 2 y = 0, y(a) = y(b) = 0 λ n = nπ b − a , y n = sin  nπ(x − a) b − a  , n ∈ N y n , y n  = b − a 2 • y  + λ 2 y = 0, y(a) = y  (b) = 0 λ n = (2n − 1)π 2(b − a) , y n = sin  (2n − 1)π(x − a) 2(b − a)  , n ∈ N y n , y n  = b − a 2 • y  + λ 2 y = 0, y  (a) = y(b) = 0 λ n = (2n − 1)π 2(b − a) , y n = cos  (2n − 1)π(x − a) 2(b − a)  , n ∈ N 2259 y n , y n  = b − a 2 • y  + λ 2 y = 0, y  (a) = y  (b) = 0 λ n = nπ b − a , y n = cos  nπ(x − a) b − a  , n = 0, 1, 2, . . . y 0 , y 0  = b − a, y n , y n  = b − a 2 for n ∈ N 2260 Appendix L Green Functions for Ordinary Differential Equations • G  + p(x)G = δ(x − ξ), G(ξ − : ξ) = 0 G(x|ξ) = exp  −  x ξ p(t) dt  H(x − ξ) • y  = 0, y(a) = y(b) = 0 G(x|ξ) = (x < − a)(x > − b) b − a • y  = 0, y(a) = y  (b) = 0 G(x|ξ) = a − x < • y  = 0, y  (a) = y(b) = 0 G(x|ξ) = x > − b 2261 • y  − c 2 y = 0, y(a) = y(b) = 0 G(x|ξ) = sinh(c(x < − a)) sinh(c(x > − b)) c sinh(c(b − a)) • y  − c 2 y = 0, y(a) = y  (b) = 0 G(x|ξ) = − sinh(c(x < − a)) cosh(c(x > − b)) c cosh(c(b − a)) • y  − c 2 y = 0, y  (a) = y(b) = 0 G(x|ξ) = cosh(c(x < − a)) sinh(c(x > − b)) c cosh(c(b − a)) • y  + c 2 y = 0, y(a) = y(b) = 0, c = npi b−a , n ∈ N G(x|ξ) = sin(c(x < − a)) sin(c(x > − b)) c sin(c(b − a)) • y  + c 2 y = 0, y(a) = y  (b) = 0, c = (2n−1)pi 2(b−a) , n ∈ N G(x|ξ) = − sin(c(x < − a)) cos(c(x > − b)) c cos(c(b − a)) • y  + c 2 y = 0, y  (a) = y(b) = 0, c = (2n−1)pi 2(b−a) , n ∈ N G(x|ξ) = cos(c(x < − a)) sin(c(x > − b)) c cos(c(b − a)) 2262 [...]... kind, 164 4 Bessel’s equation, 162 2 Bessel’s Inequality, 12 96 Bessel’s inequality, 1340 bilinear concomitant, 917 binomial coefficients, 22 76 binomial formula, 22 76 boundary value problems, 1109 branch principal, 6 branch point, 270 branches, 6 calculus of variations, 2 060 canonical forms constant coefficient equation, 10 18 of differential equations, 10 18 cardinality of a set, 3 Cartesian form, 184 Bernoulli... of, 241 contour integral, 465 convergence absolute, 5 26 Cauchy, 5 26 comparison test, 529 Gauss’ test, 5 36 in the mean, 12 96 integral test, 530 of integrals, 1470 Raabe’s test, 535 ratio test, 531 root test, 533 sequences, 525 series, 5 26 uniform, 5 36 convolution theorem and Fourier transform, 1554 for Laplace transforms, 1490 convolutions, 1490 counter-clockwise, 241 22 86 curve, 240 closed, 240 continuous,... conjugate, 182 , 184 complex derivative, 360 , 361 complex infinity, 242 complex number, 182 magnitude, 185 modulus, 185 complex numbers, 180 arithmetic, 193 set of, 3 vectors, 193 complex plane, 184 first order differential equations, 80 3 computer games, 2 280 connected region, 240 constant coefficient differential equations, 930 continuity, 53 uniform, 55 continuous piecewise, 55 continuous functions, 53, 5 36, 539... equation, 80 8 function bijective, 5 injective, 5 inverse of, 6 multi-valued, 6 single-valued, 4 surjective, 5 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, 1 562 of derivatives, 1 564 2 288 ... 361 Analytic continuation Fourier integrals, 1550 analytic continuation, 437 analytic functions, 2225 anti-derivative, 473 Argand diagram, 184 argument of a complex number, 1 86 argument theorem, 501 asymptotic expansions, 1251 integration by parts, 1 263 asymptotic relations, 1251 autonomous D.E., 992 average value theorem, 499 Bessel functions, 162 2 generating function, 162 9 of the first kind, 162 8. .. proposition theorem: a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions 1 For even more fun say it in your best Elmer Fudd accent “This next pwobwem is weawy quite twiviaw” 2 282 Appendix W whoami 2 283 Figure W.1: Graduation, June 13, 2003 2 284 Index a + i b form, 184 Abel’s formula, 910 absolute convergence, 5 26 adjoint of a differential... 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, 1 563 of derivatives, 1 564 table of, 2255 Fourier transform alternate definitions, 1544 closure relation,... degree, 774 economics, 2 280 eigenfunctions, 1330 eigenvalue problems, 1330 2 287 eigenvalues, 1330 elements of a set, 2 empty set, 2 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, 7 86 Euler-Mascheroni constant, 161 1 exact differential equations,... equations, 10 18 cardinality of a set, 3 Cartesian form, 184 Bernoulli equations, 984 2 285 Cartesian product of sets, 3 Cauchy convergence, 5 26 Cauchy principal value, 63 4, 15 48 Cauchy’s inequality, 497 Cauchy-Riemann equations, 367 , 2225 chicken spherical, 2277 clockwise, 241 closed interval, 3 closure relation and Fourier transform, 1552 discrete sets of functions, 1297 codomain, 4 comparison test, 529... exact, 782 , 945 first order, 773, 791 homogeneous, 774 homogeneous coefficient, 7 86 inhomogeneous, 774 linear, 774 order, 773 ordinary, 773 scale-invariant, 1000 separable, 780 without explicit dep on y, 9 46 differential operator linear, 902 Dirac delta function, 1041, 12 98 direction negative, 241 positive, 241 directional derivative, 157 discontinuous functions, 54, 1337 discrete derivative, 1 167 discrete . + 1 2 sinh(x − y) See Figure M.1 for plots of the hyperbolic circular functions. 2 267 -2 -1 1 2 -3 -2 -1 1 2 3 -2 -1 1 2 -1 -0.5 0.5 1 Figure M.1: cosh x, sinh x and then tanh x 2 2 68 Appendix N Bessel Functions N.1. 2y coth(x ± y) = 1 ± coth x coth y coth x ± coth y = sinh 2x ∓ sinh 2y cosh 2x − cosh 2y 2 266 Function Sum and Difference Identities sinh x ± sinh y = 2 sinh 1 2 (x ± y) cosh 1 2 (x ∓ y) cosh x +. a) b − a  , n = 0, 1, 2, . . . y 0 , y 0  = b − a, y n , y n  = b − a 2 for n ∈ N 2 260 Appendix L Green Functions for Ordinary Differential Equations • G  + p(x)G = δ(x − ξ), G(ξ − : ξ) = 0 G(x|ξ)