Justifications that ei
= cos() + i sin()
ei x = COs( x ) + i sin( x )
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
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 separation of
(1/g) dg = i dx
(1/g) dg =
ln| g | = i x + C
| g | = ei x + C = eC ei x
| g | = C2 ei x
g = C3 ei 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 ei 0 = C3
These functions are equal when C3 = 1.
COs( x ) + i sin( x ) = ei x
Justification #2: the series method
(This is the usual justification given in textbooks.)
By use of Taylor's Theorem, we can show the following to be true for all
sin x = x - x3/3! + x5/5! -
x7/7! + x9/9! - x11/11! + ...
Knowing that, we have a mechanism to determine the value of ei,
because we can express it in terms of the above series:
COs x = 1 - x2/2! + x4/4! - x6/6!
+ x8/8! - x10/10! + ...
ex = 1 + x + x2/2! + x3/3! +
x4/4! + x5/5! + x6/6! + x7/7!
+ x8/8! + x9/9! + x10/10! + x11/11!
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! +
We know how to evaluate an imaginary number raised to an integer power,
which is done as such:
i1 = i
We can see that it repeats every four terms. Knowing this, we can simplify
the above expansion:
i2 = -1 terms repeat every four
i3 = -i
i4 = 1
i5 = i
i6 = -1
e^(i) = 1 +
i - 2/2!
- i3/3! + 4/4!
+ i5/5! - 6/6!
- i7/7! + 8/8!
+ i9/9! - 10/10!
- i11/11! +
It just so happens that this power series can be broken up into two very
Now, look at the series expansions for sine and cosine. The above above
equation happens to include those two series. The above equation can therefore
be simplified to
[1 - 2/2!
+ 4/4! - 6/6!
+ 8/8! - 10/10!
[i - i3/3!
+ i5/5! - i7/7!
+ i9/9! - i11/11!
COs() + i sin()
An interesting case is when we set
= , since the above equation
i) = -1 + 0i = -1.
which can be rewritten as
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 ex series converge as indicated for imaginary