Euler's Formula, and the Exponential Form
Chapter Twenty-Three
Syllabus topic Module 1, "1.1 Complex Numbers"
Pages 54 to 55 of 303
In one line
The bracket cos t + i sin t is exactly e to the power it, so every complex number can be written r e to the power it.
The formula
Euler's formula states the identity below, for any real t.
e^(i t) = cos(t) + i sin(t)
The next chapter proves it. This one says what it is and how to use it, because the using is what the examination asks for.
Put it together with the polar form and every complex number except zero has a third way of being written.
r e^(i t) = r cos(t) + i r sin(t)
That is the exponential form, or the Euler form. MU's label calls it "Exponential form of complex numbers".
The three forms, one number
| Form | 1 + i | Written |
|---|---|---|
| Cartesian | 1 + i | x + iy |
| Polar | sqrt(2)(cos(pi/4) + i sin(pi/4)) | r(cos t + i sin t) |
| Exponential | sqrt(2) e^(i pi/4) | r e^(it) |
All three are the same object. Which you use is a matter of what you are about to do:
- adding, use Cartesian;
- multiplying, dividing, powering or rooting, use exponential;
- reading off a modulus and an angle, use polar or exponential.
Why the exponential form makes everything easy
Because the index laws do all the work. Look at what multiplication, division, powers and conjugates become.
(r e^(i a))(q e^(i b)) = r q e^(i(a + b))
(r e^(i a))/(q e^(i b)) = (r/q) e^(i(a - b))
(r e^(i a))^3 = r^3 e^(3 i a)
conjugate(r e^(i a)) = r e^(-i a)
1/(r e^(i a)) = (1/r) e^(-i a)
Every one of those is a rule you have known since school for real exponentials: add the indices, subtract the indices, multiply the index by the power. The whole of the previous four chapters, the multiplication rule, the division rule, De Moivre's theorem and the root formula, is contained in those five lines.
De Moivre's theorem, in particular, stops being a theorem and becomes an index law.
(e^(i t))^5 = e^(5 i t)
e^(5 i t) = cos(5t) + i sin(5t)
The four values everyone should know
e^(i pi/2) = i
e^(i pi) = -1
e^(3 i pi/2) = -i
e^(2 i pi) = 1
The second is Euler's identity, usually written as e to the i pi plus one equals zero, and it is the most quoted equation in mathematics. What it says is unremarkable once you have the picture: travelling half way round the unit circle from 1 brings you to minus 1.
And the periodicity, which is the source of every multiple-valued difficulty later.
e^(2 i pi) = 1
e^(i(t + 2 pi)) = e^(i t)
The complex exponential is periodic with period 2 pi i. The real exponential never repeats a value; the complex one repeats every full turn. That is the whole reason a complex number has n nth roots and infinitely many logarithms.
Euler's Formula, and the Exponential Form
Converting, in both directions
To exponential form. Find the modulus and the argument exactly as for polar form, then write r e to the i argument.
1 + i sqrt(3) = 2 e^(i pi/3)
-4 = 4 e^(i pi)
-2i = 2 e^(-i pi/2)
From exponential form. Expand with Euler's formula.
3 e^(i pi/6) = 3 sqrt(3)/2 + 3i/2
5 e^(-i pi/2) = -5i
2 e^(i pi/4) = sqrt(2) + i sqrt(2)
A worked power, to show how short it gets
Compute (1 + i) to the tenth, which the De Moivre chapter did in three lines.
1 + i = sqrt(2) e^(i pi/4)
(sqrt(2) e^(i pi/4))^10 = 32 e^(10 i pi/4)
32 e^(5 i pi/2) = 32i
Five pi over two is a full turn plus a quarter, and e to the i pi over two is i.
Where you will meet this again
In the second half of this module. The Laplace transform of a sine or a cosine is computed most easily by writing it as a combination of complex exponentials, because an exponential is the one function whose integral is itself.
cos(t) = (e^(i t) + e^(-i t))/2
sin(t) = (e^(i t) - e^(-i t))/(2i)
Those two are worth learning now. They are Euler's formula and its conjugate added and subtracted, and they are the bridge between the trigonometric functions and the exponential one. The hyperbolic functions, three chapters from here, are the same two combinations without the i.
Check yourself
e^(i pi/3) = 1/2 + i sqrt(3)/2
e^(-i pi) = -1
abs(e^(i t)) = 1
(2 e^(i pi/6))^6 = -64
e^(i pi/4) e^(i pi/4) = i
conjugate(e^(i pi/7)) = e^(-i pi/7)
The third line is worth stating in words: e to the it always has modulus one, for every real t, so e to the it traces out the unit circle as t runs from 0 to 2 pi. That single sentence is what makes the exponential form a modulus times a direction.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.