Recipe

Evaluate the complex exponential $\exp(z)$

Use when: you're given $z=x+iy$ and need $\exp(z)$ written out in $a+bi$ form

Goal: the value of $\exp(z)$ as an explicit real and imaginary part, with any full lap already accounted for in $y$

Taught in The complex exponential function

Quick recipe

Split $z$ into $x$ and $y$
Reduce $y$ to within one lap if it's larger than $2\pi$
Evaluate $\cos y$ and $\sin y$
Multiply by $e^x$ and write the result in $a+bi$ form
Sanity-check against the geometry

1
Read $x=\operatorname{Re}(z)$ and $y=\operatorname{Im}(z)$ directly off $z$ ; the definition $\exp(z)=e^x(\cos y+i\sin y)$ treats these as two separate numbers doing two separate jobs.

Example

For $z=3+i\,\dfrac{13\pi}{6}$ : $x=3$ and $y=\dfrac{13\pi}{6}$ .

Why: $x$ and $y$ feed completely different parts of the formula ( $x$ into $e^x$ , $y$ into $\cos y+i\sin y$ ), so keeping them separate from the start avoids mixing the two.
2
Compare $y$ to $2\pi$ . If it sits outside one full lap, add or subtract whole multiples of $2\pi$ until it lands in a convenient range.

Example

$\dfrac{13\pi}{6}$ is bigger than $2\pi=\dfrac{12\pi}{6}$ , so subtract one full lap: $\frac{13\pi}{6}-2\pi=\frac{13\pi}{6}-\frac{12\pi}{6}=\frac{\pi}{6}.$
- $y$ already lies within one lap (or is already a familiar angle) — skip ahead to Step 3.
- $y$ is bigger than $2\pi$ (or more negative than $-2\pi$ ) — subtract (or add) enough copies of $2\pi$ to bring it back within one lap before continuing.
Why: $\cos y$ and $\sin y$ both have period $2\pi$ , so this step never changes the final answer; it only makes evaluating the trig values easier.
3
Evaluate the cosine and sine of the (possibly reduced) angle, using the unit circle.

Example

$\cos(\pi/6)=\dfrac{\sqrt3}{2}$ and $\sin(\pi/6)=\dfrac12$ .

Why: these two numbers are exactly the real and imaginary parts of $e^{iy}$ from Euler's formula, so Step 4 only has to multiply them by $e^x$ .
4
Compute $e^x$ , then multiply both the cosine value and the sine value by it to get the real and imaginary parts.

Example

$\exp\!\left(3+i\,\frac{13\pi}{6}\right)=e^3\left(\frac{\sqrt3}{2}+\frac12 i\right)=\frac{\sqrt3\,e^3}{2}+\frac{e^3}{2}i.$

Why: $e^x$ is the same real factor multiplying both parts at once, since $e^x(\cos y+i\sin y)$ distributes across both terms identically.
5
Check that the result sits on the circle of radius $e^x$ at angle $y$ (mod $2\pi$ ) from the positive real axis, matching the picture of $\exp$ sending vertical and horizontal lines to circles and rays.

Example

$\left|\exp(3+i\,13\pi/6)\right|=e^3$ , and the angle is $\pi/6$ past the positive real axis, exactly the point $\dfrac{\sqrt3}{2}+\dfrac12 i$ scaled outward by $e^3$ .

Why: the modulus and angle of the output are exactly $e^x$ and $y$ (mod $2\pi$ ), so this check catches an arithmetic slip before it's reported as the final answer.

Common mistakes

Assuming growing $y$ makes the output bigger, the way growing $x$ does; only $x$ controls size, $y$ only controls direction.
Forgetting to reduce $y$ modulo $2\pi$ and reporting an angle where $\cos y$ and $\sin y$ aren't the standard unit-circle values you'd otherwise recognize.
Multiplying only one of $\cos y$ and $\sin y$ by $e^x$ , leaving the other term without its real factor.
Reporting $\exp(z)$ as purely real or purely imaginary just because $x$ or $y$ looks like a "nice" number, without actually evaluating both trig values.

Evaluate the complex exponential \exp(z)

Evaluate the complex exponential $\exp(z)$