Mathematical Tools for Physics
5—Fourier Series 133 This time average of the power is (up to that constant factor that I’m ignoring) 〈 f^2 〉 = lim T→∞ 1 2 T ∫+ ...
5—Fourier Series 134 In quantum mechanics, Fourier series and its generalizations will manifest themselves in displaying the dis ...
5—Fourier Series 135 Factor these complex exponentials in order to put this into a nicer form. =eiπx/L e−iπx(N+1)/L−eiπx(N+1)/L ...
5—Fourier Series 136 Problems 5.1 Do the results in Eq. ( 4 ) by explicitly calculating the integrals. 5.2 The functions with pe ...
5—Fourier Series 137 5.8 In the two problems 5 and 6 you improved the convergence by choosing boundary conditions that better ma ...
5—Fourier Series 138 5.13 For the functione−αt on 0 < t < T, express it as a Fourier series using periodic boundary condit ...
5—Fourier Series 139 5.19 In the calculation leading to Eq. ( 31 ) I assumed thatf(t)is real and then used the properties ofanth ...
5—Fourier Series 140 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 141 out in 5.32 An input potential in a circuit is given to be a square wave±V 0 at frequency ω. What is the ou ...
Vector Spaces The idea of vectors dates back to the early 1800’s, but the generality of the concept waited until Peano’s work in ...
6—Vector Spaces 143 The common example of directed line segments (arrows) in two or three dimensions fits this idea, because you ...
6—Vector Spaces 144 In axioms 1 and 2 I called these operations “functions.” Is that the right use of the word? Yes. Without goi ...
6—Vector Spaces 145 12 Like example 10, but ∑∞ 1 |ak| p<∞. (p≥ 1 ) 13 Like example 6, but ∫b adx|f(x)| p<∞. 14 Any of exam ...
6—Vector Spaces 146 B~ A~ A~+B~ A~ 2 A~ (A~+B~) +C~ A~+ (B~+C~) The associative law, axiom 3, is also illustrated in the picture ...
6—Vector Spaces 147 is of course not a vector space. 3 f f 1 2 3 f f + f = f 1 2 Function Spaces Is example 2 a vector space? Ho ...
6—Vector Spaces 148 subset and multiplying any element of the subset by a scalar leaves it in the subset. It is a “subspace.” Pr ...
6—Vector Spaces 149 Beginning with the most elementary problems in physics and mathematics, it is clear that the choice of an ap ...
6—Vector Spaces 150 Suppose you have the relation between two functions of time A−Bω+γt=βt that is, that the twofunctionsare the ...
6—Vector Spaces 151 The first of Eqs. ( 3 ) has two independent solutions, x 1 (t) =e−γtcosω′t, and x 2 (t) =e−γtsinω′t whereγ=− ...
6—Vector Spaces 152 6.5 Norms The “norm” or length of a vector is a particularly important type of function that can be defined ...
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