Fourier Transform from Fourier Series

Before you see this post,

see

Definite Integral of Multiplied 2 Trigonometric Functions

Taylor Series

Express Functions from Taylor Series

We can think f(t) is sum of wave functions and a constant.

if t=0, f(0) = a_0.

Then, let's find a_0.

We can find a_0 like process above.

Through

Definite Integral of Multiplied 2 Trigonometric Functions,

we can find a_n ( except n = 0 ) and b_n.

Use

Express Functions from Taylor Series,

we can develop the formula.

In the process, we can derive Fourier Transform from Fourier Series.

Definite Integral of Multiplied 2 Trigonometric Functions

Before you see this post,

see

Integral Some Functions

Synthesis of Trigonometric Functions

Integral Some Functions

Before you see this post,

see

e ( mathematical constant )

Differential of A Function : sin x, cos x

Relationship between Integral and Differential

Relationship between Integral and Differential

Before you see this post,

see

Riemann Sum and Riemann Integral

and

Error Estimate of Riemann Sum

f(q) can be integrated by Riemann integral and is a continuous function

and

F(x) can be differentiated.

Integral is an inverse operation of Differential.

Error Estimate of Riemann Sum

Before you see this post,

see

Riemann Sum and Riemann Integral

f(x) can be integrated by Riemann integral.

If you want to know more,

see this report.

Riemann Sum and Riemann Integral

If the limit of Riemann sum is convergence,

f(x) can be integrated by Riemann integral.

If f(x) can be integrated by Riemann integral,

some properties in picture below are valid.

If you want to know error of Riemann Sum

see

Error Estimate of Riemann Sum