Mathematical Tools for Physics
15—Fourier Analysis 453 The functiongis called* the Fourier transform off, andfis the inverse Fourier transform ofg. Examples Fo ...
15—Fourier Analysis 454 Ifx >+athen bothx+aandx−aare positive, which implies that both exponentials vanish rapidly as k→+i∞. ...
15—Fourier Analysis 455 The change of variables makes this a standard integral, Eq. (1.10), and the other factor, with the expon ...
15—Fourier Analysis 456 = ∫ dk 2 π g(k) ∫ dxf*(x)eikx = ∫ dk 2 π g(k) [∫ dxf(x)e−ikx ]* = ∫∞ −∞ dk 2 π g(k)g*(k) (9) This is Par ...
15—Fourier Analysis 457 If you hear what you think of as a single note, it will not last forever. It starts and it ends. Say it ...
15—Fourier Analysis 458 If there are several frequencies, the result is a sum. g(ω) = ∑ n Ane−(ω−ωn) (^2) /σ (^2) n ⇐⇒ f(t) = ∑ ...
15—Fourier Analysis 459 Here I’ve introduced the occasionally useful notation thatF(f)is the Fourier transform off. The boundary ...
15—Fourier Analysis 460 In the last line I interchanged the order of integration, and in the preceding line I had to use another ...
15—Fourier Analysis 461 ∫∞ −∞ dω 2 π e−iω(t−t ′) −mω^2 −ibω+k =− 2 πi ∑ ω± Res C 3 C 4 (15) The denominator in Eq. ( 14 ) is−m(ω ...
15—Fourier Analysis 462 Eq. ( 15 ) is then ∫∞ −∞ dω 2 π . e −iω(t−t′) −mω^2 −ibω+k =−i [ e−i(ω ′−iγ)(t−t′) − 2 mω′ + e−i(−ω ′−iγ ...
15—Fourier Analysis 463 In this basis, 〈 um,um 〉 =L/ 2 , so f(x) = ∑∞ n=1 〈 un,f 〉 〈 un,un 〉un(x) = 2 L ∑∞ n=1 〈 un,f 〉 un(x) No ...
15—Fourier Analysis 464 What is the sine transform of a derivative? Integrate by parts, remembering thatfhas to approach zero at ...
15—Fourier Analysis 465 Problems 15.1 Invert the Fourier transform,g, in Eq. ( 7 ). 15.2 What is the Fourier transform ofeik^0 x ...
15—Fourier Analysis 466 15.9 Derive Eq. ( 9 ) from Eq. ( 8 ). 15.10 What is the analog of Eq. ( 9 ) for two different functions? ...
15—Fourier Analysis 467 15.17 Schroedinger’s equation is −i ̄h ∂ψ ∂t =− ̄h^2 2 m ∂^2 ψ ∂x^2 +V(x)ψ Fourier transform the whole e ...
15—Fourier Analysis 468 15.23 For both the sine and cosine transforms, the original functionf(x)was defined for positivexonly. E ...
Index Abramowitz and Stegun, 9 , 412 absolute convergence, 37 acceleration, 79 Adams methods, 332 stable 335 alternating symbol, ...
Index 470 chain rule, 27, 211 , 212, 241 characteristic equation, 190, 197, 198 characteristic polynomial, 205 charged rings, 52 ...
Index 471 ∇, 218, 226, 259, 263 components 264 identities 265 ∇^2 , 268 ∇i, 271 δij, 155, 183, 204, 280 derivative, 11, 173, 409 ...
Index 472 erf, 8 , 24, 28, 348 error function, 8 , 24, 25, 58 essential singularity, 423 , 445 Euclidean fit, 338, 353 Euler: co ...
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