Euler's Formula and the Complex Exponential Basis

FS18: Euler’s Formula and the Complex Exponential Basis

In FS02 we learned to treat entire functions as vectors. In FS03 we learned that sines and cosines can be orthogonal function-space directions. FS04–FS17 then used those directions to describe periodic functions, waveforms, and their partial sums. There is now a powerful change of coordinates available: instead of keeping the sine and cosine directions separate, we can combine them using complex exponentials.

The reason is Euler’s formula:

(1)

This equation connects exponential functions, circular motion, and the trigonometric functions we have already used for Fourier series. It also makes differentiation of a harmonic almost effortless. We will derive the formula, visualize its geometry, construct the complex exponential basis, and prove its orthogonality. In FS19 we will use that orthogonality to derive the complex Fourier coefficients, while FS20 will systematically relate them to the real coefficients a_n and b_n.

1 A complex number is a two-component number

A complex number is written

(2)

where a and b are real. Its real and imaginary parts are Re z = a and Im z = b. On the complex plane, z is the point (a,b) or the arrow from the origin to that point. The magnitude and complex conjugate are

(3)

Multiplication by i rotates an arrow counterclockwise through 90^∘. For example,

(4)

The pattern repeats after four steps. This cycling will become the mechanism behind Euler’s formula.

A real physical displacement or voltage is not made imaginary by adopting complex notation. We use a complex function as an efficient mathematical representation and eventually take its real part. The imaginary part is an auxiliary quadrature component that helps us keep track of phase.

2 Derive Euler’s formula from Taylor series

Recall the three Maclaurin expansions, where z may be complex for the exponential series:

(5)

For real 𝜃, the sine and cosine series are

(6)

(7)

The exponential series converges absolutely for every complex z. Therefore we may substitute z = i𝜃 and regroup its terms without changing the sum:

(8)

Step 1: calculate successive powers of i

Use

(9)

Thus, term by term, Equation (8) becomes

ei𝜃 =	1 + i𝜃 − − i
	+ + i −.	(10)

Step 2: separate the real and imaginary terms

Collect terms with no factor i:

(11)

Collect terms carrying i:

(12)

Comparing with Equations (6) and (7), the real group is cos 𝜃 and the expression multiplying i is sin 𝜃. Consequently,

(13)

No new trigonometric assumption was needed. Euler’s formula follows from the same Taylor-series machinery used in calculus.

3 Visualizing Euler’s formula on the unit circle

In Equation (1), the real coordinate is cos 𝜃 and the imaginary coordinate is sin 𝜃. The corresponding arrow ends at (cos 𝜃, sin 𝜃). Its length is

(14)

Thus e^i𝜃 describes a point on the unit circle. Increasing 𝜃 moves it counterclockwise. The number i𝜃 is in the exponent, but the result does not grow or decay in magnitude: it rotates in phase.

Two checks anchor the geometry. At 𝜃 = 0,

(15)

At 𝜃 = π∕2,

(16)

At 𝜃 = π we obtain the celebrated special case

(17)

Complex phase and amplitude

If A is a positive real amplitude, then

(18)

Its magnitude is A, and its phase is 𝜃. A general nonzero complex number can be written in polar form

(19)

The phase is defined modulo 2π: 𝜃 and 𝜃 + 2πk describe the same direction for any integer k.

4 Conjugation gives Euler’s formula in reverse

Replace 𝜃 by −𝜃 in Euler’s formula and use cos(−𝜃) = cos 𝜃 and sin(−𝜃) = − sin 𝜃:

(20)

Taking the complex conjugate of a unit-length complex exponential therefore reverses the sign of the phase:

(21)

Add Equations (1) and (20):

(22)

Therefore

(23)

Subtract the second equation from the first:

(24)

Therefore

(25)

This is the first major connection to our Fourier-series work: each real sine or cosine harmonic can be assembled from two complex exponential harmonics with opposite signs in the exponent.

5 Positive and negative frequency are directions of rotation

Consider two rotating complex numbers,

(26)

The phase of z₊ increases with time, so it rotates counterclockwise. The phase of z₋ decreases with time, so it rotates clockwise. Both have magnitude one. Their real parts are the same, but their imaginary parts have opposite signs:

(27)

(28)

Why the negative frequencies are not new negative-energy waves

A negative frequency in this representation means an opposite phase-rotation direction. It does not mean the ordinary physical frequency meter reports a negative number, and it does not imply negative mechanical energy. A real cosine cannot distinguish the two directions by itself because its real projection is the same for both. The full complex representation retains that sign information.

Why the exponential basis is convenient for differentiation

Let ϕ(t) = e^iωt. Then

(29)

A derivative changes only the multiplying coefficient; the exponential shape is unchanged. This property is one reason complex exponentials are so useful in differential equations and linear systems. It converts differentiation of a harmonic into multiplication by iω.

6 Build the complex exponential Fourier basis

We now return to functions of the spatial variable x on [−π,π]. Define

(30)

The notation ℤ means all integers:

(31)

Unlike the real sine/cosine presentation, the complex exponential basis includes negative indices explicitly. The index n = 0 gives

(32)

Thus the constant function is included as a mode rather than written separately. For n = 1 and n = −1,

(33)

(34)

These are different complex functions even though their real parts coincide.

General interval and physical period

For a spatial interval [−L,L], whose periodic extension has length 2L, define

(35)

The permitted signed Wavenumbers are

(36)

For a time-periodic signal with period T, separately use

(37)

Here x and t are different physical variables. We obtain the time formula by substituting the mathematical half-period L = T∕2 and replacing the coordinate x by t, not by identifying physical length with physical time.

Mode counting

The set {e^inx} _n=−N^N has 2N + 1 distinct complex directions: one constant direction, N positive-frequency directions, and N negative-frequency directions. This matches the 2N + 1 real directions in

(38)

when restricted to real functions and paired appropriately. We will make the change of coefficient coordinates precise in FS20.

7 A necessary correction to the FS02 inner product

FS02 used real-valued functions and the inner product

(39)

For genuinely complex-valued functions, copying this definition without any change causes a problem. For example, the integral of e^ixe^ix over [−π,π] is zero, despite the function e^ix being nonzero. That would incorrectly give zero squared length.

The correct complex inner product (with the first argument taken to be linear) is

(40)

The overbar means complex conjugation. This is called a Hermitian inner product. Some textbooks place the conjugation on the first argument; that is an equally valid convention, provided it is used consistently. We will use Equation (40) throughout FS18–FS20.

Why conjugation matters for length

For g = f, the integrand is

(41)

Thus

(42)

It is real, nonnegative, and zero only when f is zero almost everywhere. Just as in FS11, functions that agree almost everywhere represent the same element of L².

The complex inner product obeys conjugate symmetry,

(43)

and linearity in the first entry,

(44)

In the second entry it is conjugate-linear:

(45)

This distinction is essential when projections and coefficients become complex.

8 Derive orthogonality of the complex Fourier basis

Using the scaled basis in Equation (35), calculate

(46)

For real x, conjugation changes i to −i:

(47)

Multiply the two exponential factors and combine their exponents:

(48)

Therefore

(49)

Case 1: choose two different integers n≠m

Let q = n − m. Then q is a nonzero integer. Integrate the exponential:

(50)

Evaluate the endpoints:

(51)

Because q is an integer, both endpoint exponentials equal (−1)^q. Their difference is zero. Therefore

(52)

Case 2: choose the same integer n = m

Now the exponent in Equation (49) is zero. We do not divide by n − m; instead, evaluate the constant integrand directly:

(53)

Combine both cases using the Kronecker delta δ_nm introduced in FS03:

(54)

On [−π,π] specifically, this becomes 2πδ_nm.

The normalized complex basis

Because every exponential mode has squared norm 2L, its normalized form is

(55)

Then

(56)

The factor is different from the used for the real sine and cosine modes, because the complex exponential has magnitude one at every point, rather than a squared sine or cosine whose average is 1∕2.

A common error worth checking

If the conjugate is omitted, the integral is instead

(57)

This is a legitimate integral identity, but it is not the complex inner-product orthogonality identity. It couples opposite indices instead of matching equal indices. Equations (54) and (57) should not be confused.

9 A first finite complex exponential expansion

The complex basis allows us to write a finite periodic waveform as

(58)

At this stage, think of the C_n simply as the amounts of the selected function-space directions. In FS19 we will derive their integral formula from the orthogonality result just proved.

Example: rewrite a familiar real waveform

Take the real function

(59)

Use Equations (23) and (25) on each harmonic separately:

(60)

(61)

Consequently,

f(x) =	2 + 2ieix − 2ie−ix
	+ ei2x + e−i2x.	(62)

Notice that the coefficients of the negative-index modes are the complex conjugates of those of the matching positive-index modes. That is not a coincidence.

10 Why a real signal has conjugate pairs

A real-valued waveform must be unchanged by complex conjugation:

(63)

Suppose a finite complex series has the form in Equation (58). Pair the positive and negative indices:

(64)

If

(65)

then the pair is

(66)

which is real for every x. Also, the constant coefficient C₀ must be real.

Conversely, for a real function represented by a unique complex Fourier expansion, its coefficients obey this conjugate symmetry. FS19 will derive the coefficient formula, and FS20 will prove the full equivalence with a_n and b_n.

Is a complex coefficient an amplitude?

Not quite: a complex coefficient encodes both amplitude and phase. For C_n = ρe^iα, the real pair is

(67)

Thus 2|C_n| is the amplitude of that real cosine pair for n > 0, while arg C_n supplies the phase. The zero-frequency coefficient is handled separately and is not doubled.

11 Worked physics example I: forced harmonic motion

Consider a mass-spring-damper system,

(68)

where m is mass, b is damping coefficient, and k is spring stiffness. Assume b > 0, k > 0, and a steady sinusoidal response after any transient has decayed.

The real forcing is the real part of a rotating complex exponential:

(69)

Seek the complex response

(70)

where Y is a constant but potentially complex amplitude, called a phasor. Our physical displacement is y(t) = Re z(t).

Differentiate without expanding into trigonometric functions:

(71)

(72)

Substitute into the complex version of Equation (68):

(73)

Cancel the common nonzero exponential factor:

(74)

Therefore

(75)

The magnitude of the phasor determines the displacement amplitude:

(76)

Its phase determines the timing relative to the driving force. If δ = atan2(bω,k −mω²), the steady displacement is

(77)

The arctangent with two arguments keeps the quadrant correct when k − mω² changes sign. The complex notation has condensed a differential-equation calculation into an algebraic one because e^iωt is preserved under differentiation.

12 Worked physics example II: a traveling wave

A traveling sinusoidal displacement can be written using a complex exponential:

(78)

For real positive A, Euler’s formula gives the familiar real waveform

(79)

The phase is

(80)

Track one crest by holding Φ constant. Differentiating that condition gives

(81)

Hence the crest moves at the phase velocity

(82)

for nonzero k. If both k and ω are positive in the chosen convention, it travels toward increasing x.

The same representation simplifies partial derivatives:

(83)

(84)

For the one-dimensional wave equation y_tt = c²y _xx, substitution yields

(85)

The complex exponential is useful because the wave equation reduces to a relation between frequency and wavenumber. The later FS22 and FS27 lectures will develop Standing Waves, wave packets, and dispersion in depth.

13 Connection to vector projection and Hilbert space

FS11 treated Fourier series as projections onto orthogonal function-space directions. The same interpretation applies here, but the directions are complex-valued. We use the complex Hilbert space L²([−L,L]; ℂ), equipped with the Hermitian inner product of Equation (40).

For a selected exponential mode ϕ_m, a one-direction projection has the form

(86)

with the residual r = f − p_m required to satisfy

(87)

Substituting gives

(88)

using our convention of linearity in the first argument. Thus the geometry is unchanged from FS02, except that the inner product includes conjugation.

The normalization already established in Equation (53) is

(89)

FS19 will take the final step: substitute the explicit exponential into the projection coefficient, then show that the result generates the full complex Fourier series. We intentionally defer that step so the meaning of the basis is established before the coefficient formula is introduced.

14 Common mistakes and checks

1. e^i𝜃 is not a growing real exponential. It has magnitude one and encodes a rotation.
2. e^i𝜃 and e^−i𝜃 have the same real part but different imaginary parts. They are distinct complex basis functions.
3. Negative frequency is a signed rotation rate in the complex representation. It does not mean negative energy.
4. For a real waveform, opposite-index coefficients occur as complex conjugate pairs. They are not independent real-valued amplitudes.
5. A complex inner product requires a conjugate; without it, the squared length of a nonzero function may incorrectly vanish.
6. The exponential mode norm over [−L,L] is 2L, whereas a nonconstant real sine or cosine has squared norm L.
7. Spatial wavenumber k has units of radians per unit length; temporal angular frequency ω has units of radians per second. Do not identify them merely because they appear similarly in exponents.
8. A phasor is a complex computational representation. The physically observable displacement or voltage is recovered by taking the real part.

15 Self-check questions

1. Which powers of i generate the alternating signs in the Taylor expansion of e^i𝜃?
2. Why does |e^i𝜃| = 1 for every real 𝜃?
3. Show why cos 𝜃 requires both e^i𝜃 and e^−i𝜃.
4. What does the sign of the frequency determine on a complex-plane phasor plot?
5. Why is ∫ _−π^πe^ixe^ix dx = 0 not evidence that e^ix has zero norm?
6. Compute ⟨e^i2x,e^−ix⟩ on [−π,π] with the correct complex inner product.
7. How many complex exponential modes are retained when n = −3,−2,…, 3?
8. Why must the coefficients for +n and −n be related if a finite complex Fourier sum represents a real signal?
9. What changes in the forced oscillator when a time derivative acts on e^iωt?
10. In cos(kx − ωt), why does the minus sign between kx and ωt represent travel toward increasing x when k,ω > 0?

16 Further practice

1. Use Taylor series to derive e^−i𝜃 = cos 𝜃 − i sin 𝜃 independently.
2. Rewrite 4 cos 3x + 2 sin 2x using complex exponential modes.
3. Show that e^i3x and e^−i3x are orthogonal in the complex inner product on [−π,π].
4. Derive the mode norm on a time interval of length T.
5. Verify directly that ∥e^inx∕∥ = 1 on [−π,π].
6. Show why d²e^ikx∕dx² = −k²e^ikx.
7. Find the amplitude and phase of 3 + 4i, then write it in polar exponential form.
8. Show how the complex pair Ce^inx + Ce^−inx becomes a real cosine with a phase shift.

17 What comes next

FS18 has demonstrated how the real trigonometric basis can be reorganized using rotating complex exponential functions. The modes e^inx are orthogonal in the Hermitian inner product and their differentiation is especially simple. The negative-index modes are as important as the positive-index modes for reconstructing real signals.

FS19 will now derive the complex Fourier coefficient formula by projecting a periodic function onto one chosen mode, just as we did for the real coefficients in FS04 and FS05. FS20 will show that the real and complex Fourier series carry exactly the same information. The later FS25 and FS26 lectures will interpret the discrete complex coefficients as a frequency spectrum and then pass to the Fourier-transform integral.

References