Express Functions from Taylor Series

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read

Taylor Series

e ( mathematical constant )

Differential of A Function : sin x, cos x

Taylor Series and Maclaurin Series

We can express some functions by the series.

Fourier Transform from Fourier Series

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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

1st order ODE

Before you see this post,

see

e ( mathematical constant )

Relationship between Integral and Differential

Exact Differential Equation

There is a formula of ODE like former.

We can express the formula like latter.

We can consider 2 conditions.

First, the formula is non-exact differential equation.

Second, it is exact differential equation.

If it's non-exact differential equation, we can put F(x,y) and multiply with it. 

Let's suppose F(x,y) = F(x) , x dependent function.

We can find what function y is.