Complex exponential and trigonometric functions

Try

The real exponential turns addition into multiplication. In the complex plane it also turns imaginary increments into rotation. For example exp(log 2+iπ/2)=2i: the real part determines a scale, while the imaginary part determines an angle.

Start by plotting exp(it) at t=0, π/2, π and 3π/2. Every point has modulus one, although its real and imaginary coordinates change. Moving t by a full turn returns to the same point; the complex exponential therefore cannot have a single inverse on the whole plane.

Predict

Predict whether exp(z)=exp(w) forces z=w. The values zero and 2πi provide an immediate test. Predict separately whether the exponential can equal zero. Repeated values and a missing value are different issues: a function may fail to be one-to-one without ever vanishing.

The next argument establishes both statements without relying on a finite plot of a spiral.

Experiment

Compare exact inputs rather than decimal approximations to angles.

Input	Exponential	Modulus
0	1	1
iπ	−1	1
log 3+iπ	−3	3
log 3+3iπ	−3	3

Change the real part by log 2 and check that every output length doubles. Change the imaginary part by π/2 and check the quarter-turn instead.

Understand

Define exp z by the series Σ zⁿ/n!, starting at n=0. It converges absolutely for every z, and uniformly on each bounded disk, by comparison with the real exponential series. The usual power-series differentiation theorem, proved later in this unit, gives derivative exp z. The addition law can already be proved algebraically: absolute convergence permits multiplication and regrouping by total degree, and the binomial theorem turns the coefficient sum into (z+w)ⁿ/n!. Therefore exp(z+w)=exp z exp w.

For real y, separating the even and odd terms gives exp(iy)=cos y+i sin y, with the ordinary real trigonometric series. Thus exp(x+iy)=exp x(cos y+i sin y). Its modulus is exp x. This also proves nonvanishing; alternatively exp z exp(−z)=1 does so directly.

Equality exp z=exp w is equivalent to exp(z−w)=1. The modulus forces Re(z−w)=0. The real sine and cosine values then force Im(z−w)=2πk for an integer k. These are all repetitions, not merely some examples.

Define complex cosine and sine by (exp(iz)+exp(−iz))/2 and (exp(iz)−exp(−iz))/(2i). Their familiar algebraic identities continue to hold, but bounds remembered from real inputs may fail completely.

Apply

Test the false statement “The modulus of complex cosine is at most one.” At z=iy, cosine becomes (exp y+exp(−y))/2, which grows without bound. Write the actual substitution so that the two minus signs are visible.

Next solve exp z=2−2i by separating modulus and argument. The real part must be log(2√2); the imaginary part is −π/4+2πk. Checking the full family is more informative than reporting a calculator’s principal value alone.

Continue

The logarithm lesson selects one continuous angle range so that a local inverse can be used. Optional depth: prove sin²z+cos²z=1 directly from the exponential definitions. This identity involves algebraic squares, not squared moduli; confusing those two statements would incorrectly transfer real bounds to complex inputs.