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 eᶻ 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 eˣ to the complex plane. This function is not only entire but also periodic.

To define the complex exponential function, we seek a function f: ℂ→ℂ that satisfies two key properties:

1. f(z₁ + z₂) = f(z₁)·f(z₂) for all z₁, z₂ ∈ ℂ
2. Consistency with the Real Exponential: f(x) = eˣ where x is a real number.

Let z = x + iy, where x, y ∈ ℝ. f(z) = f(x + iy) =[1] f(x)·f(iy) =[2] eˣ·f(iy)

Writing f(iy) = A(y) + iB(y), f(z) = eˣ·A(y) + ieˣ·B(y), where A(y) and B(y) are real-valued functions of y.

For f(z) to be a well-behaved function, we want it to be differentiable everywhere, which means it must satisfy the Cauchy-Riemann equations. The real and imaginary parts of f(z) are: u(x, y) = eˣ·A(y), v(x, y) = eˣ·B(y). The partial derivatives are uₓ = eˣ·A(y), u_y = eˣ·A’(y), vₓ = eˣ·B(y), and v_y = eˣ·B’(y).

Applying the Cauchy-Riemann equations, uₓ = v_y and u_y = - vₓ, we get: eˣ·A(y) = eˣ·B’(y), then, A(y) = B’(y). Similarly, eˣ·A’(y) = - eˣ·B(y), then A’(y) = -B(y).

From these two equations, we can derive a second-order differential equation for B(y): B’’(y) = -B(y) ↭ B’’(y) + B(y) = 0. This linear, homogeneous, constant-coefficient ODE has characteristic equation r² + 1 = 0 ⇒ r = ±i, the general solution is $y(t) = e^{at}(αcos(bt) + βsin(bt))$, where ‘α’ and ‘β’ are real numbers.

From A(0) = 1 and B(0) = 0 ⇒ β = 1, α = 0. Thus, we have uniquely determined the functions A(y) and B(y): B(y) = sin(y), A(y) = cos(y). Substituting these back into the expression for f(z), we arrive at the definition of the complex exponential function, f(z) = eˣ·cos(y) + ieˣ·sin(y) = eˣ(cos(y) + isin(y)).

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

Sine and Cosine functions

For a complex number z = x + i·y, (where x, y ∈ ℝ), $exp(e) = e^{x+iy} = e^x(cos(y) + isin(y))$

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

Complex Definitions of Trigonometric Functions

These relationships naturally extend to complex arguments, giving us the standard complex-analytic definitions: $cos(z) = \frac{eⁱᶻ+e⁻ⁱᶻ}{2}, sin(z) = \frac{eⁱᶻ-e⁻ⁱᶻ}{2i}$.

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

Definition. Hyperbolic sine: $\sinh z = \frac{e^z - e^{-z}}{2}$. Hyperbolic cosine: $\cosh z = \frac{e^z + e^{-z}}{2}$. These definitions are valid for all complex numbers $z \in \mathbb{C}$, and use the exponential function to build smooth, entire functions. They emerge naturally from evaluating regular trigonometric functions at imaginary inputs: $\sin(iz) = i\sinh z, \cos(iz) = \cosh z, \sinh(iz) = i\sin z, \cosh(iz) = \cos z$

Key Properties

• Differentiation: $\frac{d}{dz} \sinh z = \cosh z,\frac{d}{dz} \cosh z = \sinh z$
• Identity (hyperbolic version of the Pythagorean identity): $\cosh^2 z - \sinh^2 z = 1$.

Let’s express sin(z) and cos(z) in terms of their real and imaginary parts.

cos(z) = cos(x + iy) = $\frac{e^{i(x+iy)}+e^{-i(x+iy)}}{2} = \frac{e^{-y+xi}+e^{y-ix}}{2} = \frac{e^{-y}(cos(x)+isin(x))+e^{y}(cos(x)-isin(x))}{2} = cos(x)·\frac{e^{y}+e^{-y}}{2}-isin(x)\frac{e^{y}-e^{-y}}{2} = cos(x)cosh(y) -isin(x)sinh(y)$

sin(z) = sin(x + iy) = $\frac{e^{i(x+iy)}-e^{-i(x+iy)}}{2i} = \frac{e^{-y+xi}-e^{y-ix}}{2i} = \frac{e^{-y}(cos(x)+isin(x))-e^{y}(cos(x)-isin(x))}{2i} = sin(x)·\frac{e^{y}+e^{-y}}{2}+icos(x)\frac{e^{y}-e^{-y}}{2} = sin(x)cosh(y) +icos(h)sinh(y)$

Properties

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

This confirms that our complex definitions are consistent extensions of the real functions.

• Entire Functions. Both $\sin z$ and $\cos z$ are entire functions (holomorphic everywhere in $\mathbb{C}$). Their reciprocals (like $\tan z, \sec z$) have poles where the denominators vanish. $\tan z$ and $\sec z$ have poles where $\cos z = 0$, $\cot z$ and $\csc z$ have poles where $\sin z = 0$.

  Since the exponential function $e^z$ is entire (holomorphic on the entire complex plane), and sine and cosine are defined entirely through linear combinations of exponentials ($e^{iz}$ and $e^{-iz}$), they inherit this property. No discontinuities, no branch cuts, no singularities—just smooth, complex-analytic behavior everywhere.

• 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 $n \in \mathbb{Z}$