Math Tables: Complexity, a z e r a.

A z e r a

Math2.Org Math Tables: Complexity

Justifications that e i

= cos(

) + i sin(

)

Justification #1: from the derivative

Consider the function on the right hand side (RHS)
f(x) = cos( x ) + i sin( x )

Differentiate this function
f ' (x) = -sin( x ) + i cos( x) = i f(x)

So, this function has the property that its derivative is i times the original function.
What other type of function has this property?

A function g(x) will have this property if
dg / dx = i g
This is a differential equation that can be solved with seperation of variables
(1/g) dg = i dx

(1/g) dg =

i dx
ln| g | = i x + C
| g | = e i x + C = e C e i x
| g | = C₂ e i x
g = C₃ e i x

So we need to determine what value (if any) of the constant C₃ makes g(x) = f(x).
If we set x=0 and evaluate f(x) and g(x), we get
f(x) = cos( 0 ) + i sin( 0 ) = 1
g(x) = C₃ e i 0 = C₃
These functions are equal when C₃ = 1.

Therefore,
cos( x ) + i sin( x ) = e i x

Justification #2: the series method

(This is the usual justification given in textbooks.)

By use of Taylors Theorem, we can show the following to be true for all real numbers:

Sin x = x - x 3 /3! + x 5 /5! - x 7 /7! + x 9 /9! - x 11 /11! + .

Cos x = 1 - x 2 /2! + x 4 /4! - x 6 /6! + x 8 /8! - x 10 /10! + .

E x = 1 + x + x 2 /2! + x 3 /3! + x 4 /4! + x 5 /5! + x 6 /6! + x 7 /7! + x 8 /8! + x 9 /9! + x 10 /10! + x 11 /11! + .

E^(

i) = 1 + (

i) + (

i) 2 /2! + (

i) 3 /3! + (

i) 4 /4! + (

i) 5 /5! + (

i) 6 /6! + (

i) 7 /7! + (

i) 8 /8! + (

i) 9 /9! + (

i) 10 /10! + (

i) 11 /11! + .

I 1 = i
i 2 = -1 terms repeat every four
i 3 = -i
i 4 = 1
i 5 = i
i 6 = -1
etc.

E^(

i) = 1 +

i -

2 /2! - i

3 /3! +

4 /4! + i

5 /5! -

6 /6! - i

7 /7! +

8 /8! + i

9 /9! -

10 /10! - i

11 /11! + .

E^(

i) =
[1 -

2 /2! +

4 /4! -

6 /6! +

8 /8! -

10 /10! + . ]
+
[i

- i

3 /3! + i

5 /5! - i

7 /7! + i

9 /9! - i

11 /11! + . ]

E^(

i) = cos(

) + i sin(

)

E^(

i) = -1 + 0i = -1.

E^(

i) + 1 = 0. Special case

Which remarkably links five very fundamental constants of mathematics into one small equation.

Again, this is not necessarily a proof since we have not shown that the sin(x), cos(x), and e x series converge as indicated for imaginary numbers.

В итоге: Math2.Org Math Tables: Complexity Justifications that e i = cos( ) + i sin( ) Justification #1: from the derivative Consider the function on the right hand side (RHS) f(x) = cos( x