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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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

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Integral Some Functions

Synthesis of Trigonometric Functions

Integral Some Functions

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see

e ( mathematical constant )

Differential of A Function : sin x, cos x

Relationship between Integral and Differential

1st order ODE

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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.

Exact Differential Equation

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see

Relationship between Integral and Differential

Total differential

Definition of exact differential equation

Solutions of exact differential equation

Matrix Operation

pic. 1

Let's think about possible conditions of matrix operation

by reading pic. 1.

Example

pic. 2

pic. 3

There are n x n matrices, A and B.

Every matrix has rows and columns like pic. 3.

Let's operate matrices.

pic. 4

We can operate like pic. 4.

Example

pic. 5

Examples of Matrix Operation

Introduction to Differential Equation

Differential Equations

1. Type

Ordinary Differential Equations

ex ) y" + y' = 0 , y" + 7 x( y' ) ^3 = 3x , ...

Partial Differential Equations

ex )

, ...

2. Linearity

Linear Differential Equations 

ex) y" + xy' = 7x,  8y"' + 3y' = xy, ...

Nonlinear Differential Equations

ex) (y"")^19 + (y")^ 30 - xy = 30x, ...

3. Order

1st, 2nd, 3rd, ... , n-th Order Differential Equations

ex)

y' + y^3 = 0 -> 1st order differential equation

(x^50)y" + (y')^300 + y = x -> 2nd order differential equation

x(y"')^20 + y = 3x^398 -> 3rd order differential equation

Linearity

Definition of linearity

Function L has additivity and Homogeneity.

Let's find whether some functions have linearity.

In these example, some functions don't have linearity,

f(x) = ax, a is in real number, has linearity.

All linear functions don't have linearity.

f(x) = ax + b, b is not 0, is a linear function but it doesn't have linearity.

If functions are linearity, we can analyze and expect easily.

Let's see the graph below.

In this graph, If we want to find quantity,

 we don't have to estimate quantity for length of time of 287.

We just estimate quantity for length of time of 142 and 145.

Because we can use additivity in the graph above.

f( 142+145 ) = f( 142 ) + f( 145 )

Another method, We just estimate quantity for only length of time of 1.

Because we can also use homogeneity in the graph above.

f( 287 ) = 287 f( 1 )

If a graph doesn't have linearity, we can't expect quantity easily.

Look at this graph, 

we are unable to easily assume a function of the graph,

so, we can't easily expect quantity of length of time of 287

before that time.

Cramer's Rule

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Properties of Determinant

I've used Gauss-Jordan Elimination 

to solve simultaneous equations.

There is another method to solve simultaneous equations,

Cramer's Rule

You can understand how to solve simultaneous equations by Cramer's rule.

Zeckendorf's theorem

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Fibonacci Number

Zeckendorf's theorem states 

that every positive integer can be represented uniquely 

as the sum of one or more distinct Fibonacci numbers 

in such a way 

that the sum does not include any two consecutive Fibonacci numbers.

Vector in Coordinate System

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Eigenvalue in Matrix

Diagonalizable

Total differential

Unit Vector

pic. 1

Let formulas be supposed like that pic. 1.

pic. 2

We can express the vector v like pic. 2.

pic. 3

We can find relationship 

between total differentials in Cartesian and general coordinate system

in pic. 2 and pic. 3.

Fibonacci Number

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Diagonalizable

Gauss - Jordan Elimination