Complex Sine and Cosine Functions

An expert is a person who has made all the mistakes that can be made in a very narrow field, Niels Bohr

The Complex Exponential Function

The complex exponential function, denoted as $e^z$ or $\exp(z)$, is one of the most important functions in all of mathematics. It is the unique entire function that extends the real exponential function $e^x$ to the complex plane. This function is not only entire but also periodic.

To define the complex exponential function, we seek a function $f: \mathbb{C} \to \mathbb{C}$ that satisfies two key properties:

1. Additivity (Functional Equation): $f(z_1 + z_2) = f(z_1) \cdot f(z_2)$ for all $z_1, z_2 \in \mathbb{C}$.
2. Consistency with the Real Exponential: $f(x) = e^x$ for all $x \in \mathbb{R}$.

Let $z = x + iy$ with $x, y \in \mathbb{R}$. Using properties (1) and (2): $f(z) = f(x + iy) \overset{(1)}{=} f(x) \cdot f(iy) \overset{(2)}{=} e^x \cdot f(iy).$

Writing $f(iy) = A(y) + iB(y)$ where $A$ and $B$ are real-valued functions of $y$, $f(z) = e^x A(y) + ie^x B(y),$ so $u(x, y) = e^x A(y)$ and $v(x, y) = e^x B(y)$.

For $f$ to be analytic (differentiable everywhere), its real and imaginary parts must satisfy the Cauchy–Riemann equations.

Computing partial derivatives: $u_x = e^x A(y), \quad u_y = e^x A'(y), \quad v_x = e^x B(y), \quad v_y = e^x B'(y).$

The equation $u_x = v_y$ gives $e^x A(y) = e^x B'(y)$, hence $A(y) = B'(y) (\star)$.

Similarly, the equation $u_y = -v_x$ gives $e^x A'(y) = -e^x B(y)$, hence $A'(y) = -B(y) (\star\star)$.

Differentiating $A(y) = B'(y) (\star)$, we get $B''(y) = A'(y)$ and substituting $A'(y) = -B(y) (\star\star)$: $B''(y) = A'(y) = -B(y) \quad \Longrightarrow \quad B''(y) + B(y) = 0.$

This is a linear, homogeneous, constant-coefficient ODE with characteristic equation $r^2 + 1 = 0$, giving roots $r = \pm i$.

Recall. For complex conjugate roots $a \pm bi$, the general solution is $y(t) = e^{at}(\alpha\cos(bt) + \beta\sin(bt))$, where $\alpha, \beta \in \mathbb{R}$.

With $a = 0$ and $b = 1$ ($r = \pm i$), the general solution is: $B(y) = \alpha\cos y + \beta\sin y, \qquad A(y) =[(\star)] B'(y) = -\alpha\sin y + \beta\cos y.$

From property (2), $f(0) = e^0 = 1$. But $f(0) = e^0 A(0) + ie^0 B(0) = A(0) + iB(0)$, so $A(0) = 1 \quad \text{and} \quad B(0) = 0.$

From $A(0) = 1$: $-\alpha\sin 0 + \beta\cos 0 = \beta = 1$.

From $B(0) = 0$: $\alpha\cos 0 + \beta\sin 0 = \alpha = 0$.

Therefore $A(y) = \cos y$ and $B(y) = \sin y$, and we finally arrive at the definition of the complex exponential function: $\boxed{e^z = e^x(\cos y + i\sin y), \qquad z = x + iy.}$

The exponential function is well-defined for all $z \in \mathbb{C}$, entire, non-zero, consistent with the real exponential, periodic with period $2\pi i$ ($e^{z+2\pi i}=e^z$), satisfies $\frac{d}{dz}e^z = e^z$, and can be represented as a power series $\boxed{e^z = \sum_{n=0}^∞ \frac{z^n}{n!}}$ with infinite radius of convergence.

Sine and Cosine functions

From Euler’s Formula to Complex Definitions

For a complex number $z = x + iy$ (where $x, y \in \mathbb{R}$), the complex exponential is $exp(e) = e^{x+iy} = e^x(cos(y) + isin(y))$

Euler’s formula $e^{iy} = cos(y) + isin(y)$ leads to elegant expressions for the trigonometric functions. If we also write its complex conjugate $e^{-i y} = \cos(y) - i\sin(y),$ then by adding and subtracting these two expressions, we could isolate the real trigonometric functions and get: $\boxed{cos(y) = \frac{eⁱʸ+e⁻ⁱʸ}{2}, sin(y) = \frac{eⁱʸ-e⁻ⁱʸ}{2i}}$.

Complex Definitions of Trigonometric Functions

These relationships naturally extend to complex arguments, giving us the standard complex-analytic definitions: $\boxed{\cos z = \frac{e^{iz} + e^{-iz}}{2}, \qquad \sin z = \frac{e^{iz} - e^{-iz}}{2i}}$.valid for all $z \in \mathbb{C}$.

From these, all other trigonometric functions follow: $\tan z = \frac{\sin z}{\cos z}, \quad \cot z = \frac{\cos z}{\sin z}, \quad \sec z = \frac{1}{\cos z}, \quad \csc z = \frac{1}{\sin z}.$

Definition. The hyperbolic sine and cosine are defined for all $z \in \mathbb{C}$ by $\boxed{\sinh z = \frac{e^z - e^{-z}}{2}}, \qquad \boxed{\cosh z = \frac{e^z + e^{-z}}{2}}.$

These are entire functions (it follows from the fact that they are build from the entire exponential function) built from the same exponential building blocks as sine and cosine, but without the factor of $i$ in the exponent. These definitions are valid for all complex numbers $z \in \mathbb{C}$, and use the exponential function to build smooth, entire functions.

Bridge identities. Evaluating the circular functions at imaginary arguments (or vice versa) reveals that trigonometric and hyperbolic functions are essentially the same functions, rotated in the complex plane: $\sin(iz) = i\sinh z, \qquad \cos(iz) = \cosh z$, $\sinh(iz) = i\sin z, \qquad \cosh(iz) = \cos z$

Proof: $\sin(iz) = \dfrac{e^{i(iz)} - e^{-i(iz)}}{2i} = \dfrac{e^{-z} - e^{z}}{2i} = \dfrac{-(e^z - e^{-z})}{2i} = \dfrac{i(e^z - e^{-z})}{2} = i\sinh z.$

Key Properties

• Differentiation: $\frac{d}{dz} \sinh z = \cosh z,\frac{d}{dz} \cosh z = \sinh z$
• Fundamental identity (hyperbolic version of the Pythagorean identity): $\boxed{\cosh^2 z - \sinh^2 z = 1}$.
• Even/odd symmetry: $\cosh(-z) = \cosh z, \quad \sinh(-z) = -\sinh z$
• Separation into Real and Imaginary Parts. $\boxed{\cos(x+iy) = \cos x\,\cosh y \;-\; i\sin x\,\sinh y}$, $\boxed{\sin(x+iy) = \sin x\,\cosh y \;+\; i\cos x\,\sinh y}$.

Let’s write $\sin z$ and $\cos z$ in terms of their reasl and imaginary parts $x = \operatorname{Re} z$ and $y = \operatorname{Im} z$.

$$ \begin{aligned} \cos(x+iy) = \frac{e^{i(x+iy)} + e^{-i(x+iy)}}{2} &=\frac{e^{-y+ix} + e^{y-ix}}{2} \\[2pt] &= \frac{e^{-y}(\cos x + i\sin x) + e^{y}(\cos x - i\sin x)}{2}\\[2pt] &=\cos x \cdot \frac{e^y + e^{-y}}{2} - i\sin x \cdot \frac{e^y - e^{-y}}{2}\\[2pt] &=\cos x\cosh y - i\sin x \sinh y. \end{aligned} $$$$ \begin{aligned} \sin(x+iy) = \frac{e^{i(x+iy)} - e^{-i(x+iy)}}{2i} &=\frac{e^{-y+ix} - e^{y-ix}}{2i} \\[2pt] &= \frac{e^{-y}(\cos x + i\sin x) - e^{y}(\cos x - i\sin x)}{2i} \\[2pt] &= \sin x \cdot \frac{e^y + e^{-y}}{2} + i\cos x \cdot \frac{e^y - e^{-y}}{2} \\[2pt] &= \sin x \cosh y + i\cos x \sinh y \end{aligned} $$

Properties

• Reduction to Real Functions. When restricted to the real line ($y = 0$), since $\cosh 0 = 1$ and $\sinh 0 = 0$: $\cos(x + i\cdot 0) = \cos x \cdot 1 - i\sin x \cdot 0 = \cos x$ (the familiar real cosine), $\sin(x + i\cdot 0) = \sin x \cdot 1 + i\cos x \cdot 0 = \sin x$ (the familiar real sine).

This confirms that the complex definitions are consistent extensions of the real trigonometric functions — they agree on $\mathbb{R}$ and extend holomorphically to all of $\mathbb{C}$.

• Entire Functions. Both $\sin z$ and $\cos z$ are entire functions — holomorphic on the entire complex plane $\mathbb{C}$, with no singularities of any kind.

  Since the exponential function $e^z$ is entire, and $\sin z$ and $\cos z$ are defined through linear combinations of the entire functions $e^{iz}$ and $e^{-iz}$, they inherit this property. No discontinuities, no branch cuts, no singularities — just smooth, “well complex-analytic behavior” everywhere.

Their reciprocals and quotients, however, introduce isolated singularities (poles) where their denominators vanish. All these poles are simple (order 1. This is the case because the zeroes of sin(z) and cos(z) are simple, their derivatives are non-zero at the zeroes), and the functions are meromorphic on $\mathbb{C}$.

A meromorphic function is almost analytic everywhere, except for countably many singularities (poles) where it “blows up” in a controlled way, e.g., $\frac{1}{z}, \frac{1}{z^2}$, etc.

Function	Poles at
$\tan z = \sin z / \cos z$	$z = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}$
$\sec z = 1/\cos z$	$z = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}$
$\cot z = \cos z / \sin z$	$z = n\pi, \quad n \in \mathbb{Z}$
$\csc z = 1/\sin z$	$z = n\pi, \quad n \in \mathbb{Z}$

• Derivatives of Sine and Cosine. The derivatives of the complex sine and cosine functions can be found by differentiating their exponential definitions. Using the chain rule and the fact that $\frac{d}{dz}e^z = e^z$, we have $\frac{d}{dz}cos(z) = \frac{d}{dz}\frac{eⁱᶻ+e⁻ⁱᶻ}{2} = \frac{ieⁱᶻ-ie⁻ⁱᶻ}{2} = i\frac{eⁱᶻ-e⁻ⁱᶻ}{2} = -\frac{eⁱᶻ-e⁻ⁱᶻ}{2i} = -sin(z)$. Similarly, for the sine function: $\frac{d}{dz}sin(z) = \frac{d}{dz}\frac{eⁱᶻ-e⁻ⁱᶻ}{2i} = \frac{ieⁱᶻ+ie⁻ⁱᶻ}{2i} = \frac{eⁱᶻ+e⁻ⁱᶻ}{2} = cos(z)$. These results are identical to the familiar formulas from real calculus and extend naturally to the complex domain.

Higher-order derivatives: $\frac{d^2}{dz^2}\cos z = -\cos z, \frac{d^2}{dz^2}\sin z = -\sin z, \frac{d^4}{dz^4}\cos z = \cos z$ (returns to original function)

Iterating, we obtain the Taylor coefficients at 0: cos²ⁿ(0) = (-1)ⁿ, cos²ⁿ⁺¹(0) = 0, sin²ⁿ(0) = 0, cos²ⁿ⁺¹(0) = (-1)ⁿ

• Global Power-Series Expansions. From the above derivatives: $cos(z) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}z^{2n}, sin(z) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}z^{2n+1}$, for all z ∈ ℂ. These series converge for every z (radius of convergence =∞), providing an independent proof of entirety.

• Integration. ∫sin(z)dz = −cos(z) + C and ∫cos(z)dz = sin(z) + C.

• Trigonometric Identities in the Complex Plane.All the standard trigonometric identities that hold for real arguments also hold for complex arguments. For example, the Pythagorean identity, $cos^2(z) + sin^2(z) = (\frac{eⁱᶻ+e⁻ⁱᶻ}{2})^2 + (\frac{eⁱᶻ-e⁻ⁱᶻ}{2i})^2 = \frac{e²ⁱᶻ+ 2 + e⁻²ⁱᶻ}{4} + \frac{e²ⁱᶻ -2 +e⁻²ⁱᶻ}{-4} = \frac{e²ⁱᶻ+ 2 + e⁻²ⁱᶻ -e²ⁱᶻ + 2 - e⁻²ⁱ}{4} = \frac{4}{4} = 1$.

Angle Addition Formulas: $\sin(z_1 + z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2$, $\cos(z_1 + z_2) = \cos z_1 \cos z_2 - \sin z_1 \sin z_2$, $\sin(z_1 - z_2) = \sin z_1 \cos z_2 - \cos z_1 \sin z_2$, $\cos(z_1 - z_2) = \cos z_1 \cos z_2 + \sin z_1 \sin z_2$

Double Angle Formulas: $\sin(2z) = 2\sin z \cos z, \cos(2z) = \cos^2 z - \sin^2 z = 2\cos^2 z - 1 = 1 - 2\sin^2 z$

Half-Angle Formulas: $\cos^2(z/2) = \frac{1 + \cos z}{2}, \sin^2(z/2) = \frac{1 - \cos z}{2}$

• Periodicity and Symmetry Properties.

Periodicity Both sine and cosine are $2\pi$-periodic: $\sin(z + 2n\pi) = \sin z$ for all $n \in \mathbb{Z}, \cos(z + 2n\pi) = \cos z$ for all $n \in \mathbb{Z}$

This follows directly from the periodicity of the exponential function: $e^{i(z+2nπ)}=e^{iz}e^{i2nπ}=e^{iz}$

Phase Relationships: $\sin\left(\frac{\pi}{2} + z\right) = \cos z, \cos\left(\frac{\pi}{2} + z\right) = -\sin z, \sin\left(\frac{\pi}{2} - z\right) = \cos z, \cos\left(\frac{\pi}{2} - z\right) = \sin z$. This expresses how sine and cosine rotate into one another under a quarter-turn.

Symmetry Properties: $\sin(-z) = -\sin z$ (odd function), $\cos(-z) = \cos z$ (even function)

• Exponential growth. ∣sin(x+iy)∣² = sin²(x)cosh²(y) + cos²(x)sinh²(y) =[$\cosh^2 z - \sinh^2 z = 1$] sin²(x)(1 + sinh²(y)) + cos²(x)sinh²(y) = sin²(x) + sin²(x)sinh²(y) + cos²(x)sinh²(y) =[Factor out sinh²(y)] sin²(x) + (sin²(x) + cos²(x))sinh²(y) = sin²(x) + sinh²(y).

Now, let’s analyze the behavior as ∣y∣→∞. The hyperbolic sine function, sinh(y), is defined as: sinh(y) = $\frac{e^y - e^{-y}}{2}$.

As ∣y∣→∞, the term $e^{-|y|}$ approaches 0, so sinh(∣y∣) ≈ $\frac{e^{|y|}}{2}$. Therefore, as ∣y∣→∞: ∣sin(x+iy)∣² ≈ sin²(x) + $(\frac{e^{|y|}}{2})^2 ≈ sin²(x) + \frac{e^{2|y|}}{4}$. Since the exponential term $e^{2|y|}$ grows much faster than the bounded term sin²(x) (which is between 0 and 1), the asymptotic behavior is dominated by the hyperbolic sine term, ∣sin(x+iy)∣² ∼ $\frac{e^{2|y|}}{4}$. Taking the square root of both sides, we find the magnitude: ∣sin(x+iy)∣ ∼ $\sqrt{\frac{e^{2|y|}}{4}} = \frac{e^{|y|}}{2}$. Similarly, ∣cos(x+iy)∣ ∼ $\frac{e^{|y|}}{2}$ So complex sine and cosine grow exponentially in the imaginary direction! —even though on the real line they stay bounded between -1 and 1.

Complex sine and cosine are entire (analytic everywhere) but unbounded, unlike their real counterparts.

• Zeros of Sine and Cosine. $\sin z = 0$ when $z = n\pi$ for $n \in \mathbb{Z}$. $\cos z = 0$ when $z = \frac{\pi}{2} + n\pi$ for $n \in \mathbb{Z}$