Plancherel\'s theorem states that for \"any\" function f(a) we have F(k)eikan dk
ID: 1998697 • Letter: P
Question
Plancherel's theorem states that for "any" function f(a) we have F(k)eikan dk, where ikr dr F(k) F (k) is called the Fourier transform of f a). Note that F(k) is obtained by integrating over a, while f(a) is obtained by integrating over k. We can think of F(k) dk as the contribution to f(z) from complex exponentials e ikan/ 2T with spatial frequencies in the range from k to k dk This problem (based on a problem from the PHYS 380 textbook, Introduction to Quantum Mechanics, by David J. Griffiths) will guide you through an argument to make Plancherel's theorem seem plausible, by starting with the theory of ordinary Fourier series on a finite interval and then allowing the width of that interval to expand to infinity. a) Dirichlet's theorem states that "any" function f(z) defined on the interval I-a, +a can be expanded as a Fourier series (4) m cos an sin with sin COSExplanation / Answer
Answer:
Dirichlet's theorem says that any function f(x) on the interval [-a,+a] can be expressed as Fouries series.