Mathematical Tools for Physics - Department of Physics - University

5—Fourier Series 114 5.6 Return to Parseval When you have a periodic wave such as a musical note, you can Fourier analyze it. Th ...

5—Fourier Series 115 In quantum mechanics, Fourier series and its generalizations will manifest themselves in displaying the dis ...

5—Fourier Series 116 Now for the value offNat this point, fN ( L/2(N+ 1) ) = 4 π ∑N k=0 1 2 k+ 1 sin (2k+ 1)πL/2(N+ 1) L = 4 π ∑ ...

5—Fourier Series 117 6 Same as the preceding, show that these functions are orthogonal: e^2 iπx/L and e−^2 iπx/L, L−^14 (7 +i)x ...

5—Fourier Series 118 Problems 5.1 Get the results in Eq. (5.18) by explicitly calculating the integrals. 5.2 (a) The functions w ...

5—Fourier Series 119 5.9 (a) Use the periodic boundary conditions on−L < x <+Land basiseπinx/Lto writex^2 as a Fourier ser ...

5—Fourier Series 120 heading in the right direction. Ans:^4 /π+^1 / 2 sinωt−^8 /π ∑ neven> 0 cos(nωt)/(n (^2) −1) 5.17 For th ...

5—Fourier Series 121 5.26 Derive a Fourier series for the function f(x) = { Ax ( 0 < x < L/ 2 ) A(L−x) (L/ 2 < x < L ...

5—Fourier Series 122 5.35 (a) For the functionf(x) =x^4 , evaluate the Fourier series on the interval−L < x < Lusing perio ...

Vector Spaces . The idea of vectors dates back to the middle 1800’s, but our current understanding of the concept waited until P ...

6—Vector Spaces 124 6.2 Axioms The precise definition of a vector space is given by listing a set of axioms. For this purpose, I ...

6—Vector Spaces 125 8 Like example 5, but withn=∞. 9 Like example 8, but each vector has only a finite number of non-zero entrie ...

6—Vector Spaces 126 A~′ A~ α(A~+B~) Axioms 5 and 9 appear in this picture. Finally, axiom 10 is true because you leave the vecto ...

6—Vector Spaces 127 The integral of the right-hand side is by assumption finite, so the same must hold for the left side. This s ...

6—Vector Spaces 128 the numbers{ai}are called thecomponentsof~ain the specified basis. Note that you don’t have to talk about or ...

6—Vector Spaces 129 are functions, and as such they are elements of the vector space of example 2. All you need to do now is to ...

6—Vector Spaces 130 6.6 Scalar Product The scalar product of two vectors is a scalar valued function oftwovector variables. It c ...

6—Vector Spaces 131 For a norm, there are many possibilities: (1)‖(a 1 ,...,an)‖= √ ∑n k=1|ak| 2 (2)‖(a 1 ,...,an)‖= ∑n k=1 |ak| ...

6—Vector Spaces 132 sale is within 1000 feet of a school. If you are an attorney defending someone accused of this crime, which ...

6—Vector Spaces 133 sinπxL sin^2 πxL sin^3 πxL f To emphasize the relationship between Fourier series and the ideas of vector sp ...