Power series and the complex exponential

Lesson 8 introduced infinite series, tests for their convergence, power series and Taylor series. Power series are far more than a way to approximate functions near a point. They can be added, multiplied, differentiated and integrated term by term, which makes them a working tool: they give values of integrals that have no elementary antiderivative, solutions of differential equations that cannot be solved in closed form, and fast algorithms for computing constants such as π. Most strikingly, the series for \(e^x\), \(\cos x\) and \(\sin x\) reveal a deep connection between exponentials and trigonometry when \(x\) is allowed to be a complex number. Euler's formula, \(e^{i\theta} = \cos\theta + i\sin\theta\), underlies the phasor analysis of AC circuits, the description of vibrations and waves, signal processing and control theory. This lesson develops these tools and ends with the complex arithmetic that electrical and mechanical engineers use every day.

Radius and interval of convergence

Lesson 8 showed that a power series \(\sum c_n (x - a)^n\) converges for \(|x - a| < R\) and diverges for \(|x - a| > R\), where the radius of convergence \(R\) can be found with the ratio test. At the two endpoints, \(x = a \pm R\), anything can happen, and each endpoint must be checked separately with another test. The set of \(x\) where the series converges is the interval of convergence.

The radius of convergence has a simple explanation in terms of complex numbers, which this lesson introduces below: a power series centered at \(a\) converges out to the nearest point, possibly a complex one, where the function misbehaves. The series for \(1/(1 + x^2)\) about 0 has radius 1 even though the function is perfectly smooth for all real \(x\), because \(1 + x^2 = 0\) at the complex points \(x = \pm i\).

Building new series from known ones

Inside its interval of convergence, a power series can be manipulated like a polynomial. This lets you build series for new functions from a few known ones, without computing derivatives:

• Substitution: replacing \(x\) by \(-x^2\) in \(e^x = \sum x^n/n!\) gives \(e^{-x^2} = \sum (-1)^n x^{2n}/n!\).
• Multiplication by powers: \(x\sin x = x^2 - x^4/3! + x^6/5! - \cdots\).
• Differentiation term by term: differentiating \(\frac{1}{1 - x} = \sum x^n\) gives \(\frac{1}{(1 - x)^2} = \sum n x^{n-1}\), with the same radius of convergence.
• Integration term by term: integrating \(\frac{1}{1 + t^2} = \sum (-1)^n t^{2n}\) from 0 to \(x\) gives the arctangent series.

Similarly, integrating \(\frac{1}{1 + t} = \sum (-1)^n t^n\) gives \(\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\) for \(-1 < x \le 1\).

Integrals that have no elementary antiderivative

Lesson 2 noted that integrals such as \(\int e^{-x^2}\,dx\) and \(\int \frac{\sin x}{x}\,dx\) cannot be expressed with elementary functions. Power series provide both a definition and a practical way to compute them: expand the integrand, integrate term by term, and sum enough terms.

Series solutions of differential equations

Many differential equations that arise in engineering have solutions that are not elementary functions. One approach is to assume a solution in the form of a power series, \(y = \sum c_n x^n\), substitute it into the equation, and match the coefficients of each power of \(x\). This gives a recurrence relation for the coefficients.

The binomial series

For any real number \(p\), Newton's binomial series generalizes the binomial theorem to non-integer powers:

When \(p\) is a positive integer, the series stops and gives the familiar binomial expansion; otherwise it is infinite. It provides quick approximations such as \(\sqrt{1 + x} \approx 1 + x/2 - x^2/8\) and \(1/\sqrt{1 - x} \approx 1 + x/2 + 3x^2/8\).

Complex numbers

A complex number has the form \(z = a + bi\), where \(a\) and \(b\) are real and \(i\) satisfies \(i^2 = -1\). (Electrical engineers write \(j\) instead of \(i\), because \(i\) denotes current.) The real part is \(a\) and the imaginary part \(b\). Complex numbers add component by component and multiply using \(i^2 = -1\): \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\). The complex conjugate of \(z\) is \(\bar{z} = a - bi\), and \(z\bar{z} = a^2 + b^2\), which is used to divide: \(\dfrac{1}{a + bi} = \dfrac{a - bi}{a^2 + b^2}\).

A complex number can be pictured as a point in the complex plane, with the real part horizontal and the imaginary part vertical. Its distance from the origin is the modulus \(|z| = \sqrt{a^2 + b^2}\), and its angle from the positive real axis is the argument \(\theta\), so that \(z = |z|(\cos\theta + i\sin\theta)\), the polar form of Lesson 7 in a new setting.

Every polynomial equation has solutions among the complex numbers (the fundamental theorem of algebra), so complex numbers make the quadratic formula always work: \(x^2 + 2x + 5 = 0\) has the roots \(-1 \pm 2i\). In engineering, complex roots of characteristic equations signal oscillation, as MATH-201 shows.

Euler's formula

Substituting \(i\theta\) into the exponential series and separating real and imaginary parts, using \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\):

The two brackets are the series for \(\cos\theta\) and \(\sin\theta\), giving Euler's formula:

With \(\theta = \pi\) it gives \(e^{i\pi} + 1 = 0\), linking five of the most important numbers in mathematics. More usefully, every complex number can be written compactly as \(z = |z|e^{i\theta}\), and the laws of exponents then make multiplication simple: moduli multiply and arguments add, \(r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2e^{i(\theta_1 + \theta_2)}\). Multiplying by \(e^{i\phi}\) rotates a point by angle \(\phi\) about the origin. Raising to powers gives De Moivre's formula, \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\), from which multiple-angle identities follow in one line.

Euler's formula also expresses the trigonometric functions as exponentials, \(\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\) and \(\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\), the complex counterparts of the definitions of \(\cosh\) and \(\sinh\). Integrals such as \(\int e^{ax}\cos bx\,dx\), which need integration by parts twice (Lesson 1), become a single step: take the real part of \(\int e^{(a + ib)x}\,dx = \frac{e^{(a + ib)x}}{a + ib}\).

Phasors and oscillations

A sinusoidal quantity \(A\cos(\omega t + \phi)\) is the real part of \(Ae^{i\phi}e^{i\omega t}\). The complex number \(Ae^{i\phi}\), which captures the amplitude and phase, is called a phasor. Because differentiating \(e^{i\omega t}\) simply multiplies it by \(i\omega\), linear differential equations driven by sinusoids become algebraic equations for phasors. This is the basis of AC circuit analysis: a resistor, inductor and capacitor have impedances \(R\), \(i\omega L\) and \(1/(i\omega C)\), which combine like resistances.

Complex exponents also describe damped oscillations. If \(s = \sigma + i\omega\), then \(e^{st} = e^{\sigma t}(\cos\omega t + i\sin\omega t)\): the real part \(\sigma\) sets the growth or decay of the envelope and the imaginary part \(\omega\) the frequency of oscillation. A vibrating structure, a suspension or a control loop whose characteristic equation has roots with negative real parts settles down; roots with positive real parts mean growing oscillations and instability. This single idea links the series of this lesson to the vibration, circuit and control courses that follow.