Euler’s Formula: Rotations, Powers, and Roots

1. Derive Euler’s formula from the series

Substitute $z=i\theta$ into the exponential series, with $\theta$ real. The powers of $i$ cycle through $i,-1,-i,1$. Separate the real terms from the imaginary terms.

The real terms are exactly the cosine series. The imaginary terms are $i$ times the sine series. The exponential series is absolutely convergent, so this regrouping is valid.

The result is Euler’s formula. At $\theta=\pi$, it gives $e^{i\pi}=-1$, often written $e^{i\pi}+1=0$.

Remember: For real $\theta$, $e^{i\theta}$ is a point on the unit circle at angle $\theta$.

2. Write a complex number as r e^(iθ)

Euler’s formula shortens the polar form to $z=re^{i\theta}$. For nonzero $z$, $r=|z|>0$ and $\theta$ is any argument of $z$.

For a general exponent $a+ib$ with real $a,b$, the exponential law gives $e^{a+ib}=e^a(\cos b+i\sin b)$. The real part of the exponent controls the modulus; the imaginary part controls the angle.

Remember: Rectangular, polar, and exponential forms describe the same number in different useful ways.

3. Multiplication scales and rotates

The exponential law makes multiplication especially simple: multiply the moduli and add the angles. For division by a nonzero number, divide the moduli and subtract the angles.

Multiplying by $i=e^{i\pi/2}$ rotates a point anticlockwise by $90^\circ$ without changing its distance from the origin. More generally, multiplication by $re^{i\theta}$ scales distances by $r$ and rotates by $\theta$.

Angles add as arguments, but the sum may lie outside the principal range. Add or subtract full turns if a principal argument is required.

Remember: Multiplication combines lengths by multiplying and directions by adding angles.

4. Powers: De Moivre’s theorem

Multiplying a number by itself repeats the same scaling and rotation. For a positive integer $n$, raise the modulus to the $n$th power and multiply the angle by $n$.

This gives De Moivre’s theorem. It also holds for zero and negative integer powers when the base is nonzero. Fractional powers require more care because there are generally several roots.

Remember: For integer powers, polar or exponential form avoids expanding large brackets.

5. Find every nth root

To solve $w^n=z$ for nonzero $z=re^{i\theta}$ and a positive integer $n$, write $w=\rho e^{i\phi}$. Comparing moduli gives $\rho^n=r$. Comparing angles gives $n\phi=\theta+2k\pi$.

There are exactly $n$ distinct roots. Take $k=0,1,\ldots,n-1$; further integer values repeat the same points. The roots lie equally spaced around a circle of radius $r^{1/n}$. For $z=0$, the only distinct root is $w=0$.

Remember: An nth-root problem for a nonzero complex number has $n$ equally spaced solutions.

6. Recover sine and cosine from exponentials

Apply Euler’s formula to $\theta$ and $-\theta$. Adding the equations isolates cosine; subtracting them isolates sine. These identities also hold for complex inputs $z$, using their power-series definitions.

Where $\cos z\ne0$, tangent is defined as $\tan z=\sin z/\cos z$. Sine and cosine are defined for all complex inputs, but tangent is undefined where its denominator vanishes.

To see why this is useful, multiply $e^{i\alpha}$ and $e^{i\beta}$ and expand their sine-cosine forms. Equating real and imaginary parts gives the familiar angle-addition identities.

Remember: Complex exponentials turn trigonometric identities into algebra.

7. Application: represent a wave by a rotating number

A real oscillation can be written as $x(t)=A\cos(\omega t+\phi)$, where $A$ is a nonnegative amplitude, $\omega$ is angular frequency in radians per unit time, and $\phi$ is a phase angle. Euler’s formula writes this as the real part of a complex exponential.

The constant complex number $X=Ae^{i\phi}$ is called a phasor. The time-dependent expression $Xe^{i\omega t}$ rotates; its real part is the measurable oscillation. A phasor keeps track of amplitude and phase together.

For signals with the same angular frequency, add the phasors and then take the real part. This can be easier than combining trigonometric expressions. If you know calculus, differentiation multiplies $Xe^{i\omega t}$ by $i\omega$, which explains its usefulness in differential equations.

Remember: A phasor packages amplitude and phase; take the real part to return to the real signal.

8. Application: a simple AC circuit

In sinusoidal steady state at one angular frequency $\omega>0$, complex impedance relates voltage and current phasors: $V=ZI$. A resistor contributes $R$, an inductor contributes $i\omega L$, and a capacitor contributes $1/(i\omega C)=-i/(\omega C)$. Series impedances add.

Here $R$ is resistance in ohms, $L$ is inductance in henries, and $C$ is capacitance in farads. Electrical engineering often writes $j$ instead of $i$ to avoid confusion with current; the mathematical rule is the same, $j^2=-1$.

Use one consistent amplitude convention. In this example, all phasor moduli are peak amplitudes and the real signal is obtained with $e^{i\omega t}$.

Remember: In AC analysis, the modulus gives an amplitude ratio and the argument gives a phase shift.

9. A final formula sheet and study routine

Use rectangular form for addition and subtraction; exponential form for multiplication, division, powers, and roots. Check a result by converting back or substituting it into the original equation.

For roots, list every required value of $k$. For angles, keep radians and the principal-argument convention consistent. For series, check convergence before using an infinite-sum formula. These habits are more useful than memorising formulas without their conditions.

Remember: You can now connect complex arithmetic, geometry, series, trigonometry, and simple applied models.