The remarkable formula that carries Euler's name looks like this (for any real x):
e^(i·x) = cos(x)+i·sin(x)
This formula can be considered as a foundation for a definition of this concept. However, a lot of considerations were made to present it in this way and this is the only way to define complex exponentiation to preserve all the properties of exponentiation we know from using only real numbers as exponents.
Let's recall a few facts from the material presented earlier.
In Algebra - Limits chapter we discussed a number e and derived a formula that represented this number as a limit:
e = lim[n→∞] (1+1/n)^n.
We have also derived there a more general formula
e^x = lim[n→∞] (1+x/n)^n.
This is not a rigorous proof of the Euler's formula, but reasonable considerations that help us to properly define the operation of raising a real number into complex power.
Let's look at the representation of e^x as a limit above and use it formally for x=i.
e^i = lim[n→∞] (1+i/n)^n.
There is nothing undefined in this formula - we know how to add, subtract, multiply, divide and raise into power complex numbers.
In theory, if we skip the proof that this limit exists, which is beyond the scope of this lecture, this might be a proper definition of a complex exponent. But we will go further to arrive at more constructive Euler's formula.
Let's consider a base of the above expression for ei as a limit - the term (1+i/n) - and represent it as a point A on a complex plane with origin at point O where real part of a number is an abscissa and imaginary part is an ordinate of this point. Abscissa of our point equals to 1, its ordinate equals to 1/n, which in case of large n puts our point A(1,1/n) just slightly above the point P(1,0) lying on the X-axis. This "slightly above" is diminishing as n→∞, so point A moves down closer and closer to point P.
Let's use the representation of complex numbers in polar form as r·[cos(φ)+i·sin(φ)].
In this case
φ = ∠POA.
As n→∞ and, as we mentioned, point A moving down closer and closer to point P, r→1 (the length of OP) and φ→0.
We know that for angles tending to 0 there is a very important limit
lim[φ→0] sin(φ)/φ = 1
Also cos(φ) gets closer and closer to 1 as φ→0.
The above considerations prompt us to replace point (1,1/n) with a point (cos(1/n),sin(1/n)), which is infinitely close to it.
The reason for such substitution is the fact that we have to raise an expression (1+1/n) to n-th power, and this is much easier to do in the polar form. So, we substitute (1+i/n)^n with [cos(1/n)+i·sin(1/n)]^n.
The substitution we made leads us to a very interesting formula:
e^i = lim[n→∞] (1+i/n)^n ≅ [cos(1/n)+i·sin(1/n)]^n
Using the properties of polar representation of complex numbers, we can simplify the last expression:
[cos(1/n)+i·sin(1/n)]^n = cos(n·1/n)+i·sin(n·1/n) = cos(1)+i·sin(1)
(just a reminder, angles are in radians, so angle of 1 means 1 radian).
Our final formula is:
e^i = cos(1)+i·sin(1)
Let us emphasize again, we have not proven this formula, since there was no definition of complex exponentiation, but we came up with it using reasonable transformation of properties of complex numbers and exponentiation of real numbers onto the field of complex numbers.
Immediate consequence of the above formula, again transforming known properties of exponentiation of real numbers onto complex numbers and properties of polar representation of complex numbers, is for any real x:
e^(i·x) = (e^i)^x = [cos(1)+i·sin(1)]^x = cos(1·x)+i·sin(1·x) =
which is the Euler's formula we presented above.
The beauty of this formula can be compared with the beauty of a famous Einstein's formula for full energy E=m·c². Euler has brought together complex analysis, trigonometry and limits, seemingly unrelated concepts, and Einstein has brought together energy, mass and speed of light, also quite unrelated from the surface concepts.
Everything is reasonably interrelated in our Universe. We just have to learn to see these relations.
Let's now define complex exponentiation using the above formula. If our definition is correct, all the properties of exponentiation that involves only real numbers must be held in complex case, and that is something that we can prove using our definition and known properties.
We define raising of one particular real number e to any complex power a+i·b (a and b are any real numbers) as
e^(a+i·b) = e^a·e^(i·b) = e^a·[cos(b)+i·sin(b)]
Then, using the logarithms with a base e (natural logarithms), we can define this operation for any other real number. Assume we want to raise a real number d to complex power a+i·b.
We can always represent
d = e^ln(d)
where ln(d)=log[e](d) - natural (based e) logarithm of d.
d^(a+i·b) = d^a·d^(i·b) = d^a·[e^ln(d)]i·b = d^a·e^[i·b·ln(d)]
The last expression in polar form is