If \( f(P) \) and \( g(P) \) are compler. Fhurier of \( F(x) \) and \( G(x) \) respedie (a) \( 1 / 2 \pi \int_{-\infty}^{x} f(P) g(P) d P=\int_{-\infty} f(x)(x(x) d x \) (b) \( 1 / 2 \pi \int_{-\infty}^{\infty}|f(p)|^{2} d p=\int_{-\infty}^{n}|F(x)|^{2} \)
Added by Julian B.
Close
Step 1
- We are given two functions \( f(P) \) and \( g(P) \) which are the Fourier transforms of \( F(x) \) and \( G(x) \) respectively. - We need to verify two integral identities involving these functions. Show more…
Show all steps
Your feedback will help us improve your experience
Satish Yadav and 54 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
A general form of Parseval's theorem says that if two functions are expanded in Fourie? series $$ \begin{aligned} &f(x)=\frac{1}{2} a_{0}+\sum_{1}^{x} a_{n} \cos n x+\sum_{1}^{\infty} b_{n} \sin n x \\ &g(x)=\frac{1}{2} a_{0}^{\prime}+\sum_{1}^{\infty} a_{n}^{\prime} \cos n x+\sum_{i}^{\infty} b_{n}^{\prime} \sin n x \end{aligned} $$ then the average value of $f(x) g(x)$ is $\frac{1}{4} a_{0} a_{0}^{\prime}+\frac{1}{2} \sum_{1}^{\infty} a_{n} a_{n}^{\prime}+\frac{1}{2} \sum_{1}^{\infty} b_{n} b_{n}^{\prime}$. Prove this.
FOURIER SERIES
Parseval's theorem
Mengchun C.
Prove that if the functions $f_{1}, f_{2}, \ldots, f_{n}$ are continuous on a closed interval $[a, b]$ and if $k_{1}, k_{2}, \ldots, k_{n}$ are constants, then $$ \begin{array}{l} \int_{a}^{b}\left[k_{1} f_{1}(x)+k_{2} f_{2}(x)+\cdots+k_{n} f_{n}(x)\right] d x \\ \quad=k_{1} \int_{a}^{b} f_{1}(x) d x+k_{2} \int_{a}^{b} f_{2}(x) d x+\cdots+k_{n} \int_{a}^{b} f_{n}(x) d x \end{array} $$
The Integral
Properties of the Definite Integral
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Watch the video solution with this free unlock.
EMAIL
PASSWORD