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show that Assuming that a function f satisfies ∞ f(x) = % +Σ( n=1 an COS (T) + b₂ sin ¹ (TTT)), — ſ ² ƒ(z) ² dx = ³ + Σ ( 2² +6²) n=1 (1) which is known as Parseval's Theorem, and provides a relationship between the aver- age value of f (the integral on the left) and its Fourier coefficients. HINT: Multiply (1) by f and integrate between -L and L and use formulas you know. If you don't exactly remember the formulas, think about how we derived them: take equation (1) and multiply by an appropriate sine or cosine, then integrate, and use orthogonality properties of sine and cosine.

show that Assuming that a function f satisfies ∞ f(x) = % +Σ( n=1 an COS (T) + b₂ sin ¹ (TTT)), — ſ ² ƒ(z) ² dx = ³ + Σ ( 2² +6²) n=1 (1) which is known as Parseval's Theorem, and provides a relationship between the aver- age value of f (the integral on the left) and its Fourier coefficients. HINT: Multiply (1) by f and integrate between -L and L and use formulas you know. If you don't exactly remember the formulas, think about how we derived them: take equation (1) and multiply by an appropriate sine or cosine, then integrate, and use orthogonality properties of sine and cosine.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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show that Assuming that a function f satisfies ∞ f(x) = 2% +Σ( n=1 an COS (T) + b₂ sin ¹ (TTT)), — ſ ² ƒ(z) ² dx = ³ + Σ ( 2² +6²) n=1 (1) which is known as Parseval's Theorem, and provides a relationship between the aver- age value of f (the integral on the left) and its Fourier coefficients. HINT: Multiply (1) by f and integrate between -L and L and use formulas you know. If you don't exactly remember the formulas, think about how we derived them: take equation (1) and multiply by an appropriate sine or cosine, then integrate, and use orthogonality properties of sine and cosine.
Transcribed Image Text:show that Assuming that a function f satisfies ∞ f(x) = 2% +Σ( n=1 an COS (T) + b₂ sin ¹ (TTT)), — ſ ² ƒ(z) ² dx = ³ + Σ ( 2² +6²) n=1 (1) which is known as Parseval's Theorem, and provides a relationship between the aver- age value of f (the integral on the left) and its Fourier coefficients. HINT: Multiply (1) by f and integrate between -L and L and use formulas you know. If you don't exactly remember the formulas, think about how we derived them: take equation (1) and multiply by an appropriate sine or cosine, then integrate, and use orthogonality properties of sine and cosine.
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