Complex exponential

The complex exponential is the function

$e^z=e^{x+iy}=e^x(\cos y+i\sin y).$

So $u(x,y)=e^x\cos y$ and $v(x,y)=e^x\sin y$. The modulus depends only on the real part ($|e^z|=e^x$), and the argument depends only on the imaginary part ($\arg (e^z)=y\mathrm{mod}2\pi$).

The complex exponential extends the real exponential analytically to all of $\mathbb{C}$, agreeing with $e^x$ on the real axis. It is the foundation for the complex sine and cosine, logarithm, and complex powers — all built directly from $e^z$.

Properties

• Entire. Analytic on all of $\mathbb{C}$. C–R equations hold everywhere with continuous partials.
• Own derivative: $\frac{d}{dz}{e}^z=e^z$.
• Multiplicative: $e^{z_1+z_2}=e^{z_1}\cdot e^{z_2}$.
• $|e^z|=e^x$ depends only on real part.
• $\arg (e^z)=y(\mathrm{mod}2\pi )$.
• Periodic with period $2\pi i$: $e^{z+2\pi i}=e^z$.

The periodicity is striking — there is no analog in real calculus. The complex exponential repeats vertically along the imaginary direction, period $2\pi$. This is the geometric reason why the Complex logarithm is multi-valued.

Never zero

$|e^z|=e^x>0$ for all $z\in \mathbb{C}$. The equation $e^z=0$ has no solutions. In real analysis, $e^x>0$ is just a positivity statement; in complex analysis, ”$e^z\ne 0$ ever” is what blocks $0$ from the range and forces the logarithm to be defined on $\mathbb{C}\setminus \{0\}$.

Mapping behavior

• Vertical line $\mathrm{Re}(z)=a$: maps to circle $|w|=e^a$ in the $w$-plane, traversed infinitely many times as $y$ runs over $\mathbb{R}$.
• Horizontal line $\mathrm{Im}(z)=b$: maps to a ray from the origin at angle $b$.
• Horizontal strip $0\le \mathrm{Im}(z)<2\pi$ of width $2\pi$: maps bijectively onto $\mathbb{C}\setminus \{0\}$.

The “strip-to-punctured-plane” bijection is the geometric content of the complex exponential. The corresponding inverse map (any one branch of the Complex logarithm) goes from a slit plane back to a horizontal strip.

Connection to Euler’s formula

Euler’s formula $e^{i\theta }=\cos \theta +i\sin \theta$ is the specialization $z=i\theta$. Rigorously, $e^z$ is defined for general $z$ via the Taylor series

$e^z={\sum }_{n=0}^{\infty }\frac{z^n}{n!},$

which converges absolutely on all of $\mathbb{C}$. Substituting $z=i\theta$ and separating real and imaginary parts by parity of $n$:

$e^{i\theta }={\sum }_{n=0}^{\infty }\frac{(i\theta {)}^n}{n!}={\sum }_{k=0}^{\infty }\frac{(-1{)}^k{\theta }^{2k}}{(2k)!}+i{\sum }_{k=0}^{\infty }\frac{(-1{)}^k{\theta }^{2k+1}}{(2k+1)!}=\cos \theta +i\sin \theta ,$

recognizing the cosine and sine series. So Euler’s formula is the statement that the $e^z$ series, evaluated at $i\theta$, splits exactly into the cosine and sine series.

In context

• Phasors in EE use $e^{j\omega t}$ as the rotating complex exponential whose real part is $\cos (\omega t)$. See Phasor transform.
• Fourier transforms use $e^{-j\omega t}$ as a basis for representing arbitrary signals as superpositions of complex sinusoids.
• LTI systems have $e^{st}$ as eigenfunctions: feed in $e^{s_{0}t}$, get out $H(s_0)\cdot e^{s_{0}t}$. The complex exponential is the natural building block for Laplace and Fourier analysis.

In the rest of complex analysis, $e^z$ shows up in:

• Complex sine and cosine: defined as combinations of $e^{iz}$ and $e^{-iz}$.
• Complex hyperbolic functions: as combinations of $e^z$ and $e^{-z}$.
• Complex power $z^c$ via $z^c=e^{c\log z}$.
• The integrand $e^{iax}$ in computing real improper integrals via Jordan’s lemma.