MTH647 Quiz Modules 60-124

Fourier Transform of Unit Step Function:

Find the Fourier series of

For given problem, 𝑎

−𝑎

−𝑎 𝑎

Divide and multiply by 2,

2𝑖

2𝑖 ” we have

It is the required function when 𝛼 ≠0. If we put 𝛼= 0 in equation (1), we get 𝑎

−𝑎

Fourier Transform of Exponential Functions:

Fourier function of

Solution: As we know from previous lecture,

For given problem,

𝑒 2𝑝 2𝑝

2𝑝 + 𝛼2 ⇒𝑒 𝛼2 2𝑝

Let

𝑝 By putting in above relation, we have As we know (from previous lectures)

By putting, we have

By using inverse Fourier transformation,

Applying "𝐹" on both sides, we get

Examples:

Do Your Self.

Solving Integral Equation by Convolution:

Solution: For the given integral equation, taking fourier transformation on both sides,

Convolution of last function,

So, above relation can be writtena as

Let By putting equation (1) becomes

By solving we have

Taking inverse fourier transformation, So the above equation becomes,

2𝜋

Fourier Sine and Cosing Tranforms:

Example:

As we know

For given function, it can be written as

… (1)

Using integral formula, which is given as

So, by applying integral formula, equation (1) becomes

By applying limits, we get 𝑚