MATH 473: EULER’S FORMULA
Here is Euler’s Formula
Proposition 1. If x is a real number, and i =
√
−1, then
eix = cos x + i sin x.
Idea of proof. The Taylor series of the exponential function ez is still
valid if z is a complex number. We will also need the formula for powers
of i:
i0 = 1,
i1 = i,
i2 = −1,
i3 = −i,
i4 = 1,
i5 = i,
i6 = −1,
i7 = −i,
Note the formula is periodic in the exponent with period 4, so that
in+4 = in i4 = in · 1 = in .
So compute the Taylor series of eix :
(ix)2 (ix)3 (ix)4 (ix)5 (ix)6 (ix)7
+
+
+
+
+
+ ···
2
3!
4!
5!
6!
7!
x3 x4
x5 x6
x7
x2
= 1 + ix −
−i +
+i −
− i + ···
2
3!
4! 5! 6!
7!
x2 x4 x6
x3 x5 x7
=
1−
+
−
+ ··· + i x −
+
−
+ ···
2
4!
6!
3!
5!
7!
= cos x + i sin x,
eix = 1 + ix +
where the last line follows by recognizing the Taylor series for cos x and
sin x.
Euler’s formula is useful in terms of deriving more difficult trigonometric formulas from easier formulas for the exponential function. To
proceed, note that cos x is the real part of the complex number eix and
sin x is the imaginary part of the complex number eix . We can write
cos x = Re(eix ),
sin x = Im(eix ).
We can use these formulas to derive some of the standard trigonometric formulas.
Lemma 2. cos(α + β) = cos α cos β − sin α sin β.
1
i8 = 1 . . .
MATH 473: EULER’S FORMULA
2
Proof. Compute
cos(α + β) = Re(ei(α+β) )
= Re(eiα+iβ )
=
=
=
=
=
Re(eiα eiβ )
Re(cos α + i sin α)(cos β + i sin β)
Re(cos α cos β + i sin α cos β + i cos α sin β + i2 sin α sin β)
Re(cos α cos β + i sin α cos β + i cos α sin β − sin α sin β)
cos α cos β − sin α sin β.
Note the same proof shows, by consider the imaginary part of ei(α+β) ,
that
sin(α + β) = cos α sin β + sin α cos β.
To prove more trigonometric formulas, it is useful to have more a
more explicit characterization of the real and imaginary parts of eix .
First of all, note that
e−ix = cos x − i sin x.
This follows by Euler’s Formula and that cos x is an even function and
sin x is an odd function. Then we can easily compute from Euler’s
Formula
cos x = 21 eix + 12 e−ix ,
sin x = − 2i eix + 2i e−ix .
These allow us to prove some basic formulas about the integrals and
derivatives of trigonometric functions. For example we can verify the
formula for the derivative of cos x by using the formula for the derivative
of ex , Euler’s Formula, and the chain rule:
d
Proposition 3. dx
cos x = − sin x.
Proof. Compute
d
d 1 ix 1 −ix
cos x =
( e + 2e )
dx
dx 2
d
d
= 12 eix + 12 e−ix
dx
dx
1 ix
1 −ix
= 2 e · i + 2 e · (−i)
= −(− 2i eix + 2i e−ix )
= − sin x.
MATH 473: EULER’S FORMULA
3
Finally, we compute an example of the integral formulas used in the
orthogonality relations of trigonometric polynomials:
Proposition 4. Let m, n be distinct positive integers. Then
Z π
cos mx cos nx dx = 0.
−π
Proof. Since m and n are distinct and positive, we see that n − m and
n + m are both nonzero. Compute
Z π
Z π
1 imx
cos mx cos nx dx =
(e
+ e−imx ) 12 (einx + e−inx ) dx
2
−π
−π
Z π
1
= 4
(ei(m+n)x + ei(m−n)x + ei(n−m)x + e−i(m+n)x ) dx
−π
π
i(m+n)x
ei(m−n)x
ei(n−m)x
e−i(m+n)x e
1
= 4
+
+
+
i(m + n) i(m − n) i(n − m) −i(m + n) −π
Now for each integer k, Euler’s formula shows that the function eikx =
cos kx + i sin kx is periodic, so that eikx = eik(x+2π) . In particular,
π
eikx = eikπ − e−ikπ = cos(kπ) + i sin(kπ) − cos(kπ) + i sin(kπ) = 0.
−π
Apply this to the above formula with k = m + n, m − n, n − m,
− (m + n) to complete the proof.
To recap, here are the basic formulas:
eix = cos x + i sin x,
cos x = Re(eix ) = 21 eix + 12 e−ix ,
sin x = Im(eix ) = − 2i eix + 2i e−ix .