Complex sine and cosine

The complex sine and cosine extend the real sine and cosine to all of $\mathbb{C}$, defined via Euler’s formula / the Complex exponential:

$\sin z=\frac{e^{iz}-e^{-iz}}{2i},\cos z=\frac{e^{iz}+e^{-iz}}{2}.$

Both are entire (analytic on all of $\mathbb{C}$) — combinations of the entire function $e^z$.

Derivatives and identities

Standard rules carry over:

$\frac{d}{dz}\sin z=\cos z,\frac{d}{dz}\cos z=-\sin z.$

${\sin }^{2}z+{\cos }^{2}z=1.$

Sum and difference formulas, double-angle formulas — all unchanged from the real case. Plug in complex $z$ wherever you’d plug in real $x$.

Real-imaginary decomposition

Using $\sin (x+iy)=\sin x\cos (iy)+\cos x\sin (iy)$ together with $\cos (iy)=\cosh y$ and $\sin (iy)=i\sinh y$ (proved by substitution into the $e^z$ definitions):

$\sin z=\sin x\cosh y+i\cos x\sinh y$

$\cos z=\cos x\cosh y-i\sin x\sinh y.$

Memorize these — they’re how you compute $\sin (1+i)$ or $\cos (2-3i)$ in practice. The $\cosh y$ and $\sinh y$ factors are what cause the unboundedness.

Unbounded on $\mathbb{C}$

A crucial difference from the real case: $\sin z$ and $\cos z$ are unbounded on $\mathbb{C}$. Along the imaginary axis $z=iy$:

$\cos (iy)=\cosh y,\sin (iy)=i\sinh y.$

As $|y|\to \infty$, both $\cosh y$ and $|\sinh y|$ blow up exponentially. So $|\cos z|$ and $|\sin z|$ grow without bound on horizontal strips of large $|\mathrm{Im}|$.

This is exactly why $\sin z$ and $\cos z$ are non-constant despite being entire — the contrapositive of Liouville’s theorem, which says bounded entire functions are constant. Sin and cos are entire but not bounded, so they get to be non-constant.

Zeros

The zeros are the same as in the real case:

• $\sin z=0$ iff $z=n\pi$, $n\in \mathbb{Z}$.
• $\cos z=0$ iff $z=(n+1/2)\pi$, $n\in \mathbb{Z}$.

Proof: $\sin z=(e^{iz}-e^{-iz})/(2i)=0$ iff $e^{iz}=e^{-iz}$ iff $e^{2iz}=1$ iff $2iz=2\pi in$ iff $z=n\pi$. Same for cosine.

So complex sine and cosine have only real zeros — even though their values elsewhere are complex.

In context

Complex sine and cosine show up:

• In contour integrals: ${\int }_{-\infty }^{\infty }f(x)\cos (ax)dx$ is handled by replacing $\cos (ax)\to e^{iax}$ and taking the real part, then closing the contour in the upper half-plane (using Jordan’s lemma).
• As entire-but-unbounded examples illustrating Liouville’s theorem.
• In differential equations whose solutions are sinusoidal; the complex form unifies cosine-and-sine-shifted solutions.

Relationship to hyperbolic functions

Substituting $z\mapsto iz$ swaps sin/cos with sinh/cosh:

$\sin (iz)=i\sinh z,\cos (iz)=\cosh z.$

So $\sin$ and $\cos$ along the imaginary axis become $\sinh$ and $\cosh$ along the real axis — and vice versa. The two function pairs are really one pair, rotated $90°$. See Complex hyperbolic functions.