Trigonometric and Hyperbolic Functions in Complex Analysis

students, this lesson explores how familiar trigonometric and hyperbolic functions extend into the complex plane 🌍➡️$? Wait no. $c? Let's write this properly with math. Complex analysis lets us study functions like $$\sin$ z$, $$\cos$ z$, $$\sinh$ z$, and $$\cosh$ z$ when the input $z$ is complex. These functions are not just extensions of high school formulas; they reveal deep links between geometry, periodicity, exponentials, and analytic behavior.

Introduction: Why these functions matter

The trigonometric and hyperbolic functions are central examples of elementary analytic functions. They are built from the complex exponential function $e^z$, which is itself one of the most important functions in complex analysis.

By the end of this lesson, students, you should be able to:

explain what the complex trigonometric and hyperbolic functions are,
use formulas connecting them to $e^z$,
recognize their main properties, including analyticity and periodicity,
and understand why branch issues matter for inverse functions like $\arcsin z$ and $\operatorname{arcosh} z$.

A key idea is this: in complex analysis, many functions become easier to understand when written using exponentials. For example, the ordinary trigonometric functions are defined by

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

These formulas work for every complex number $z$. That means the functions are entire, which means they are analytic everywhere in the complex plane. ✨

Trigonometric functions as analytic functions

In real calculus, $\sin x$ and $\cos x$ come from angles on the unit circle. In complex analysis, we keep the same names, but define them using exponentials so that the formulas continue to make sense for complex inputs.

From the definitions

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

we can derive many familiar identities. For instance,

$\sin^2$ z + $\cos^2$ z = 1

still holds for all complex $z$. The proof is algebraic, using the exponential formulas.

The derivatives also look just like the real-variable versions:

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

Because these derivatives exist for every complex $z$, both functions are entire. That is an important point in complex analysis: differentiability in the complex sense is much stronger than differentiability in real calculus.

Example: values at imaginary inputs

Suppose $z = iy$ where $y$ is real. Then

$\sin($iy) = $\frac{e^{-y} - e^y}{2i}$ = i\,$\sinh$ y,

and

$\cos($iy) = $\frac{e^{-y} + e^y}{2}$ = $\cosh$ y.

This shows a deep connection between trigonometric and hyperbolic functions. Imaginary inputs turn circular functions into hyperbolic ones. That relationship is one of the most beautiful patterns in the subject. 🌟

Hyperbolic functions and their exponential form

The hyperbolic functions are defined using the same exponential building blocks:

$\sinh$ z = $\frac{e^z - e^{-z}}{2}$, \qquad $\cosh$ z = $\frac{e^z + e^{-z}}{2}$.

They are also entire functions, because they are combinations of entire functions.

Just as $\sin$ and $\cos$ satisfy a circular identity, $\sinh$ and $\cosh$ satisfy the hyperbolic identity

$\cosh^2$ z - $\sinh^2$ z = 1.

Notice the sign change. This mirrors the geometry of the hyperbola $x^2 - y^2 = 1$, which is where the name “hyperbolic” comes from.

Their derivatives are

$\frac{d}{dz}$$\sinh$ z = $\cosh$ z, \qquad $\frac{d}{dz}$$\cosh$ z = $\sinh$ z.

These formulas resemble the trigonometric case, except there is no minus sign in the second derivative of $\cosh z$.

Example: growth behavior

For real $x$, the function $\cosh x$ grows rapidly as $|x|$ increases because it contains $e^x$ and $e^{-x}$. In contrast, $\cos x$ stays bounded on the real axis. This difference matters in applications. For example, hyperbolic functions appear in models of hanging cables, temperature distributions, and special curves called catenaries.

In complex analysis, though, both trigonometric and hyperbolic functions are entire. Their differences are seen more in periodicity and growth than in analyticity.

Relationships, periodicity, and geometry

The trigonometric functions are periodic:

$\sin($z + $2\pi)$ = $\sin$ z, \qquad $\cos($z + $2\pi)$ = $\cos$ z.

This remains true for complex $z$. The period $2\pi$ comes from the exponential function since $e^{i(z+2\pi)} = e^{iz}$.

The hyperbolic functions are not periodic on the real axis. However, they are related to the trigonometric functions through imaginary arguments:

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

This means one family can often be translated into the other by multiplying the input by $i$.

Another useful connection is Euler’s formula:

e^{iz} = $\cos$ z + i$\sin$ z.

This formula explains why trigonometric functions are naturally tied to rotations in the complex plane. The exponential $e^{iz}$ moves around the unit circle when $z$ is real, which is why sine and cosine encode circular motion.

Example: solving a simple equation

Consider $\sin z = 0$. Using the periodicity and the exponential form, the solutions are

z = n$\pi$, \qquad n $\in$ \mathbb{Z}.

For $\cos z = 0$, the solutions are

z = $\frac{\pi}{2}$ + n$\pi$, \qquad n $\in$ \mathbb{Z}.

These are the same families of zeros as in real calculus, but now they are understood as complex solutions too.

Branch issues and inverse functions

Inverse trigonometric and inverse hyperbolic functions are more subtle. The reason is that $\sin z$, $\cos z$, $\sinh z$, and $\cosh z$ are not one-to-one on the complex plane. A single value of the output can come from many inputs.

For example, since $\sin z$ is periodic, the equation $\sin z = w$ usually has infinitely many solutions. So a true single-valued inverse does not exist on all of $\mathbb{C}$. Instead, we define branches of inverse functions by restricting the domain or choosing a cut in the complex plane.

A branch is a single-valued analytic choice of a multivalued inverse on some region. This idea is essential in complex analysis because logarithms, roots, inverse trig functions, and inverse hyperbolic functions often require branch cuts to avoid contradictions.

For instance, one formula for the inverse sine is

$\arcsin z = -i\,\log\!\left(iz + \sqrt{1-z^2}\right),$

where $\log$ and $\sqrt{\,\cdot\,}$ must be chosen with branches. Different branch choices give different values.

Similarly, an inverse hyperbolic cosine can be written as

$\operatorname{arcosh} z = \log\!\left(z + \sqrt{z-1}\,\sqrt{z+1}\right),$

again with branch choices. The square roots and logarithm are not globally single-valued on $\mathbb{C}$, so branch cuts are required.

Example: why branches matter

The complex logarithm is multivalued because

$\log$ z = $\ln|$z| + i($\arg$ z + $2\pi$ k), \qquad k $\in$ \mathbb{Z}.

Since inverse trigonometric formulas use $\log z$, they inherit this multivalued behavior. If you try to define $\arcsin z$ everywhere without a branch cut, you will not get a single analytic function.

This is a major theme of elementary analytic functions: direct formulas may be simple, but turning them into globally consistent analytic functions often requires careful domain choices.

How these functions fit into elementary analytic functions

The topic of elementary analytic functions includes $e^z$, $\log z$, powers like $z^\alpha$, and the trigonometric and hyperbolic families. students, the key unifying idea is that many of these functions are constructed from the exponential function and then extended to complex variables.

The trigonometric functions are entire because they are built from exponentials by addition, subtraction, and division by constants. The hyperbolic functions are also entire for the same reason. Their inverse functions are more delicate and require branch choices because they involve the logarithm and square root.

So when you study trigonometric and hyperbolic functions in complex analysis, you are really studying three big ideas together:

how analytic functions can be defined from $e^z$,
how periodicity and symmetry appear in the complex plane,
and how branch issues arise in inverses.

These ideas reappear throughout the rest of complex analysis, including contour integration, residues, and conformal mapping.

Conclusion

Trigonometric and hyperbolic functions are among the most important examples of elementary analytic functions. Defined through exponentials, they are entire, satisfy elegant identities, and are closely connected by imaginary arguments. Their inverse functions introduce branch issues, which are a central feature of complex analysis.

If you remember just one big idea, students, let it be this: in complex analysis, $\sin z$, $\cos z$, $\sinh z$, and $\cosh z$ are not separate worlds. They are different faces of the exponential function, and understanding that connection makes many later topics much easier. ✅

Study Notes

$\sin z$ and $\cos z$ are defined by

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

$\sinh z$ and $\cosh z$ are defined by

$\sinh$ z = $\frac{e^z - e^{-z}}{2}$, \qquad $\cosh$ z = $\frac{e^z + e^{-z}}{2}$.

All four functions are entire.
Important identities:

$\sin^2$ z + $\cos^2$ z = 1, \qquad $\cosh^2$ z - $\sinh^2$ z = 1.

Derivatives:

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

Imaginary arguments connect the two families:

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

Trigonometric functions are periodic with period $2\pi$.
Hyperbolic functions are not periodic on the real axis.
Inverse functions like $\arcsin z$ and $\operatorname{arcosh} z$ are multivalued unless branches are chosen.
Branch cuts are needed because the complex logarithm and square root are multivalued.
These functions are fundamental examples of elementary analytic functions in complex analysis.