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Fourier Transform Sine Wave

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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




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