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Fourier Transform Sine Wave Fourier Transform Sine Wave For evaluating Fourier transform sine wave of a given function under certain interval, we use following steps – Step 1: As we know that Fourier transform sine wave of a function under certain interval is f (x) =∑n=1infinity bn sin (nπx)/L So, for evaluating the sum, we use integration: F [f (x)] = √(2/π) ∫ab f(x) sin (n x) dx. Where a and b are lower and upper limits. Step 2: Now we put value of f(x), which is given in question, and the values of lower and upper limits. We have a function f(x) = x under (0, π) interval, So Fourier transform sine wave of this function is F [f (x)] = √ (2/π) ∫0π x. sin (n x) dx. Know More About Associative Property of Real Numbers Worksheets Step 3: After above two steps, we perform integration operation and evaluate the value of Fourier sine transform of a function. Now let us take an example to understand the procedure to evaluating the value of Fourier transform sine wave of a function – Example: Evaluate the Fourier sine transform for the given function f (x) = 7(π + x) and the interval is (0, π). Solution: We use following steps for evaluating the value of Fourier sine transform of a function – Step 1: We know that Fourier series for the sine under interval (0, π) is, f (x) =∑n=1infinity bn sin (nπx)/L, F [f (x)] = √(2/π) ∫0π f(x) sin (n x) dx. Step 2: Now we put the value of given function, which is f (x) = 7(π + x) , (0, π) F [f (x)] = √ (2/π) ∫0π 7(π + x) sin (n x) d x Step 3: Now we perform integration operation - F [f (x)] =- √ (2/π) ∫0π 7(π + x) (sin (n x) d x) (by using integration by parts rule) Learn More About Associative Property of Real Numbers Worksheet F [f (x)] =- √ (2/π) [7(π + x) cos (n x)/n |0π - ∫0π 7(π + x) (-cos (n x) d x], F [f (x)] =- √ (2/π) [7(π + x) cos (n x)/n |0π - ∫0π 7π (-sin (n x)/n d x], F [f (x)] =- √ (2/π) [7(π + x) cos (n x)/n |0π - 7π ∫0π (-sin (n x)/n d x], F [f (x)] =- √ (2/π) [7(π + x) cos (n x)/n - 7π (cos (n x)/n2] |0π , F [f (x)] =- √ (2/π) [7(π + π) cos (n π)/n - 7π (cos (n π)/n2 -7 π + 7 π] F [f (x)] =- √ (2/π) [7(π + π) cos (n π)/n - 7π (cos (n π)/n2] As we know that cos n π = (-1) n, so on F [f (x)] =- √2 [14(-1)n/n+ 7(-1) n/n2 is the value of Fourier transform sine wave of a function. Thank You TutorCircle.com

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