Express Functions from Taylor Series

Before read this post,

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

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

Differential of A Function : sin x, cos x

Before you see this post,

see

A Property of Limit of A Function : f(x)/g(x)

A Property of Limit of A Function : f(x)g(x)

Synthesis of Trigonometric Functions

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.

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

Differential Function : [f(x)]^r = r[f(x)]^(r-1)

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see

Differential Function : ln x

and

Differential of A Function : f(x)g(x)

q isn't zero.

Differential of A Function : f(x) / g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)

f(x) and g(x) are continuous functions.

Differential of A Function : f(x)g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)

f(x) and g(x) are a continuous functions

Differential of A Function : f(x)+g(x)

Before you see this post,

see

A Property of Limit of A Function : f(x)+g(x)

f(x) and g(x) are a continuous functions.

Green's Theorem

1. Setting

[pic 1]

Let a closed curve C be set on x,y plane.

Domain D is covered by closed curve C.

[pic 2]

Let's functions, M(x,y) and N(x,y) exist and they have partial derivatives on D.

The formula can be notated like it in [pic 2] on C.

[pic 3]

This formula can be divided like [pic 3] 

because M(x,y)dx and N(x,y)dy are on same closed curve C. 

2. Induction

2-1. Considering The Formula at X Coordinate

First, we consider the formula in [pic 2] at x coordinate.

[pic 4]

[pic 5]

We set a domain like a formula in [pic 5].

[pic 6]

So, result is like [pic 6] on x coordinate.

2-2. Considering The Formula at Y Coordinate

Next, we also consider the formula in [pic 2] at y coordinate.

[pic 7]

[pic 8]

We set a domain like a formula in [pic 8].

[pic 9]

So, result is like [pic 6] on y coordinate.

3. Conclusion

[pic 10]

We can combine same domain.

Therefore, we can prove Green's Theorem.