Transcendental Functions

Mathematics is the language in which God has written the universe, Galileo Galilei

Complex exponential

Let z = x + iy with $x, y \in \mathbb{R}$. The exponential function exp: $\mathbb{C} → \mathbb{C}, z ↦ e^z$ may be defined in several equivalent ways. These definitions agree with one another and extend the usual real exponential function to the complex plane.

1. Euler–based definition, $e^z = e^{x+iy} = e^x(\cos(y) + i\sin(y))$
2. Limit definition, e^z = $\lim_{n \to \infin}(1+\frac{z}{n})^n$
3. Power series definition, e^z = $\sum_{n=0}^{\infty} \frac{z^n}{n!}$. TThe power series converges absolutely and uniformly on every bounded subset of $\mathbb{C}$, which guarantees excellent analytic properties.

Key Properties:

1. Domain: $Dom(e^z) = \mathbb{C}$.
2. Range: $Range(e^z) = \mathbb{C} \setminus \{0\}$. For any complex number z, it can take any complex value except for zero. This means that (i) the complex exponential function never vanishes; (ii) moreover, for any non-zero complex number w ($w \in \mathbb{C} \setminus \{0\}$), there exists at least one complex number z ($z \in \mathbb{C}$) such that $e^z = w$ (In fact, infinitely many such z exist due to periodicity).
3. Agreement with the real exponential. It coincides with the real exponential function $e^z = e^x$ when z is real (that is, y = 0), so the complex exponential extends the real exponential function.
4. Entire function. It is an entire function, meaning it is holomorphic (complex-differentiable) at every point in the complex plane. Its derivative satisfies $\frac{d}{dz}e^z = e^z$.

   An entire function in complex analysis is a function that is holomorphic (complex differentiable) at every point in the complex plane (ℂ). Essentially, it’s a complex function that is smooth and well-behaved everywhere in the complex plane.

5. Euler’s Formula. The exponential definition originates from $e^{iy} = \cos(y) + i\sin(y)$. Indeed, $e^z = e^{x+iy}=e^x \cdot e^{iy} = e^x(\cos(y) + i\sin(y))$
6. Periodicity. The complex exponential is periodic in the imaginary direction: $e^{z + 2\pi i} = e^z$. Thus, its period is $2\pi i$.
7. Modulus and argument. Let $z = x + iy \in \mathbb{C}$, then $|e^z| = |e^x|\cdot |e^{iy}| =[\text{eˡʸ is a point on the unit circle in the complex plane}] |e^x| = e^x = e^{\Re(z)}$.
   For a complex number z, the exponential function is defined as $e^x\cdot e^{iy}$. Since $e^x$ is a positive real number, its argument is 0 (modulo 2π). The factor $e^{iy} = \cos(y) + i\sin(y)$ lies on the unit circle, and its argument is exactly y modulo 2π. When multiplying two complex numbers, their arguments add. Thus, $\arg(e^z) = arg(e^x) + arg(e^{iy}) = 0 + y = y~ (\text{mod } 2\pi)$.
   This also reflects the periodicity of the complex exponential $e^{z + 2\pi i} = e^z$. Geometrically:
   $e^x$ determines the distance from the origin,
   y determines the angle in the complex plane.
8. Exponential laws. $\forall z_1, z_2 \in \mathbb{C}, e^{z_1 + z_2} = e^{z_1} \cdot e^{z_2}$, $(e^z)^n = e^{nz}$ for $n \in \mathbb{Z}$
9. The map $z = x + iy \to e^z$ sends vertical lines (x = constant) to circles centered at the origin with radius $e^x$ and horizontal lines (y = constant) to rays emanating from the origin at angle y.

The Complex Logarithm

The logarithm is defined as the inverse of the complex exponential function. Since $e^z$ is periodic with period $2\pi i, e^{z + 2\pi i} = e^z$, the logarithm in the complex plane cannot be single-valued. Instead, it is a multi-valued function.

Definition (Multi-valued Logarithm). For any non-zero complex number $z \in \mathbb{C}^*$, the (multi-valued) complex logarithm is defined by log(z) = ln(|z|) + i * arg(z) where:

• |z| is the modulus (absolute value) of z,
• ln(|z|) is the natural logarithm of the modulus,
• arg(z) is the argument of z, representing the angle between the positive real axis and the vector from the origin to z.

It’s a multi-valued function due to the periodic nature of the argument (arg(z) can take multiple values differing by 2πk, where k is any integer $arg(z) = \theta + 2\pi k, k \in \mathbb{Z}$ where $\theta$ is any angle satisfying $z = |z|(\cos(\theta) + i\sin(\theta))$). Hence, the logarithm has infinitely many values: $log(z) = ln(|z|) + i(\theta + 2\pi k), k \in \mathbb{Z}$, e.g, for z = 1 + i: $log(z) = ln(\sqrt{2}) + i * (π/4 + 2πk)$. This illustrates that each nonzero complex number has infinitely many logarithms, differing by integer multiples of 2πi.

Principal Branch of the Logarithm. To obtain a single-valued logarithm, we must restrict the argument to a specific interval. For a complex number z = x + yi, the principal branch of the complex logarithm is defined as: Log(z) = ln(|z|) + i * Arg(z) $\forall z \in \mathbb{C}^*$ except along the negative real axis (where the argument jumps discontinuously) where:

• ln(|z|) is the natural logarithm of the magnitude of z
• |z| = $\sqrt{(x² + y²)}$ is the modulus (absolute value) of z
• Arg(z) is the principal argument (angle) of z, which is the angle between the positive real axis and the line from the origin to the point z in the complex plane. The value of Arg(z) is constrained within the range (−π, π].

Why the Discontinuity Occurs

Consider a point on the negative real axis, e.g., z = −r with r > 0. Its principal argument is Arg(−r) = π.

1. If we approach this point from above the negative real axis (i.e., z = −r + iε with ε > 0 small), the argument is slightly less than π, tending to π as ε→$0^+$.
2. If we approach this point from below the negative real axis (i.e., z = −r - iε with ε > 0 small), the argument is slightly greater than -π, tending to -π as ε→$0^-$.
3. Thus, as we cross the negative real axis, the argument jumps abruptly from nearly −π (just below) to π (just above), a jump of 2π.

This discontinuity is inherent to the choice of the principal branch and is why the negative real axis is called the branch cut of Log(z)..

Properties

1. Dom(Log(z)) = ℂ*, meaning it encompasses all complex numbers except for zero. This is because the logarithm is undefined at z = 0. More precisely, Log is continuous and holomorphic on $\mathbb{C} \setminus (-\infty, 0]$.
2. Consistency with the exponential. This definition ensures that $exp(Log(z)) = z, z \ne 0$.
3. Single-valued. The restriction $Arg(z) \in (-\pi, \pi]$ ensures that Log(z) assigns exactly one value to each z in its domain, e.g., z = 1 + i, $|z| = \sqrt{1² + 1²} = \sqrt{2}, Arg(z) = arctan(1/1) = π/4, Log(z) = ln(\sqrt{2}) + i * (π/4)$

Complex Trigonometric Functions

The complex trigonometric functions are defined naturally using the complex exponential function. These definitions extend the real trigonometric functions and preserve their fundamental identities.

For any $z \in \mathbb{C}$, $\sin(z) := \frac{e^{iz} - e^{-iz}}{2i}$, $\cos(z) := \frac{e^{iz} + e^{-iz}}{2}$. These are often referred to as Euler’s formulas for complex sine and cosine. They are not merely analogies: they define the complex trigonometric functions.

Properties

• Domain: $\sin(z), \cos(z): \mathbb{C} \to \mathbb{C}$.
• Range: Both functions take values in all of $\mathbb{C}$.
• Zeros: $\sin(z) = 0 \iff z = n\pi, n \in \mathbb{Z}$. $\cos(z) = 0 \iff z = \frac{\pi}{2} + n\pi, n \in \mathbb{Z}$.
• Unboundedness: Unlike real trigonometric functions, complex $\sin z$ and $\cos z$ are unbounded, e.g., $|\sin(iy)| = |\sinh(y)| \to \infty \text{ as } |y| \to \infty$
• Explicit Formulas for sin(x + iy) and cos(x+iy) $\sin(x + iy) = \sin(x) \cosh(y) + i \cos(x) \sinh(y),\cos(x + iy) = \cos(x) \cosh(y) - i \sin(x) \sinh(y)$. These formulas clearly separate the real and imaginary parts, revealing how trigonometric and hyperbolic functions are intrinsically linked in the complex plane.

Let z = x + iy, then iz = i(x+iy) = ix - y, so $e^{iz} = e^{-y + ix} = e^{-y} e^{ix} = e^{-y}(cos(x) + i sin(x))$. Similarly, $e^{-iz} = e^{-ix + y} = e^{y} e^{-ix} = e^{y}(cos(x) - i sin(x))$.

Substituting into the sine formula:

$$ \begin{aligned} sin(z) &=[\text{By definition}] \frac{e^{iz} - e^{-iz}}{2i} \\[2pt] &= \frac{e^{-y}(cos(x) + i sin(x)) - e^{y}(cos(x) - i sin(x))}{2i} \\[2pt] &=\frac{(e^{-y} - e^{y}) cos(x) + i(e^{-y} + e^{y}) sin(x)}{2i} =\frac{-(e^{y} - e^{-y}) cos(x) + i (e^{-y} + e^{y}) sin(x)}{2i} \\[2pt] &=\frac{-(2 sinh(y)) cos(x) + i (2 cosh(y)) sin(x)}{2i} = \frac{-(2 sinh(y)) cos(x)}{2i} + \frac{i (2 cosh(y)) sin(x)}{2i} \\[2pt] &[\text{Since 1/i = -i, so -1/i = i}]=i sinh(y) cos (x) + cosh(y) sin(x). \end{aligned} $$

Similarly,

$$ \begin{aligned} cos(z) &=[\text{By definition}] \frac{e^{iz} + e^{-iz}}{2} \\[2pt] &= \frac{e^{-y}(cos(x) + i sin(x)) + e^{y}(cos(x) - i sin(x))}{2} \\[2pt] &=\frac{(e^{-y} + e^{y}) cos(x) + i(e^{-y} - e^{y}) sin(x)}{2} \\[2pt] &=\frac{(2 cosh(y)) cos(x) + i (-2 sinh(y)) sin(x)}{2} \\[2pt] &=cosh(y) cos(x) - i sinh(y) sin(x). \end{aligned} $$

Complex Hyperbolic Functions

The complex hyperbolic functions are defined analogously: $\sinh(z) := \frac{e^z - e^{-z}}{2}, \quad \cosh(z) := \frac{e^z + e^{-z}}{2}$

Relationships Between Trigonometric and Hyperbolic Functions: $\sin(iz) = i\sinh z, \quad \cos(iz) = \cosh z, \sinh(iz) = i\sin z, \quad \cosh(iz) = \cos z$.

$$ \begin{aligned} sinh(z) &=[\text{By definition}] \frac{1}{2}(e^{z}-e^{-z}) \\[2pt] &= \frac{1}{2}(e^{x+iy}-e^{-(x+iy)}) = \frac{1}{2}[(e^{x}e^{iy})-(e^{-x}e^{-iy})] \\[2pt] &=[\text{Using Euler's formula}] \frac{1}{2}[e^{x}cos(y)+e^{x}isin(y)-cos(y)e^{-x}+isin(y)e^{-x}] \\[2pt] &= cos(y)(\frac{1}{2}[e^{x}-e^{-x}]) + sin(y)(\frac{i}{2}[e^{x}+e^{-x}]) = \frac{1}{2}[e^{x}-e^{-x}]cos(y)+i\frac{1}{2}[e^{x}+e^{-x}]sin(y) \\[2pt] &=sinh(x)cos(y)+icosh(x)sin(y). \end{aligned} $$$$ \begin{aligned} cosh(z) &=[\text{By definition}] \frac{1}{2}(e^{z}+e^{-z}) \\[2pt] &= \frac{1}{2}(e^{x+iy}+e^{-(x+iy)}) = \frac{1}{2}[(e^{x}e^{iy})+(e^{-x}e^{-iy})] \\[2pt] &=[\text{Using Euler's formula}] \frac{1}{2}[e^{x}cos(y)+e^{x}isin(y)+cos(y)e^{-x}-isin(y)e^{-x}] \\[2pt] &= cos(y)(\frac{1}{2}[e^{x}+e^{-x}]) + sin(y)(\frac{i}{2}[e^{x}-e^{-x}]) = \frac{1}{2}[e^{x}+e^{-x}]cos(y)+i\frac{1}{2}[e^{x}-e^{-x}]sin(y) \\[2pt] &=cosh(x)cos(y)+isinh(x)sin(y). \end{aligned} $$

$\boxed{sinh(z)=sinh(x)cos(y)+icosh(x)sin(y)}$

Special cases

• Real input (y = 0), $sinh(z) = sinh(x)cos(y)+icosh(x)sin(y) = sinh(x)\cdot 1+icosh(x)\cdot 0 = sinh(x)$. This last mathematical statement clearly demonstrates how the complex hyperbolic sine function reduces to the real hyperbolic sine function for real arguments.
• Purely imaginary input (x = 0): $sinh(z) = sinh(iy) = sinh(0)cos(y)+icosh(0)sin(y) =[\text{Since sinh(0) = 0 and cosh(0) = 1, this simplifies to}] isin(y)$. Therefore, $\boxed{sinh(z) = isin(y)}$

$\boxed{cosh(z) = cosh(x)cos(y)+isinh(x)sin(y)}$

Special cases

• Real input (y = 0), $cosh(z) = cosh(x)cos(y)+isinh(x)sin(y) = cosh(x)\cdot 1+isinh(x)\cdot0 = cosh(x)$. This last mathematical statement clearly demonstrates how the complex hyperbolic cosine function reduces to the real hyperbolic cosine function for real arguments.
• Purely imaginary input (x = 0): $cosh(z) = cosh(iy) = cosh(x)cos(y)+isinh(x)sin(y) =[\text{Since sinh(0) = 0 and cosh(0) = 1, this simplifies to}] cos(y)$, $\boxed{cosh(iy) = cos(y)}$. The hyperbolic cosine of a purely imaginary number equals the cosine of its imaginary part.

Other Trigonometric Functions

The remaining functions are defined in the usual way: $tan(z) = \frac{sin(z)}{cos(z)}, sec(z) = \frac{1}{cos(z)}$

Identities (All Hold for Complex Arguments)

• $cos²(z)+sin²(z) = 1$
• $sin(z_1+z_2) = sin(z_1)cos(z_2) + cos(z_1)sin(z_2)$
• $cos(z_1+z_2) = cos(z_1)cos(z_2) - sin(z_1)sin(z_2)$
• $cos(iz) = \frac{1}{2}(e^{i(iz)}+e^{-i(iz)}) = \frac{1}{2}(e^{-z}+e^{z}) = cosh(z)$
• $sin(iz) = \frac{1}{2i}(e^{i(iz)}-e^{-i(iz)})= \frac{1}{2i}(e^{-z}-e^{z}) = [\frac{1}{i} = -i] \frac{1}{2}i(e^{z}-e^{-z}) = isinh(z)$

Trigonometric complex functions are unbounded

The addition formulae may be written also as sin(x + iy) =[$sin(z_1+z_2) = sin(z_1)cos(z_2) + cos(z_1)sin(z_2)$] sin(x)cos(iy) + cos(x)sin(iy) = sin(x)cosh(y)+icos(x)sinh(y).

Then, |sin(x + iy)| = $\sqrt{sin²(x)cosh²(y)+cos²(x)sinh²(y)}$ = [cos²(x) + sin²(x) = 1] $\sqrt{sin²(x)cosh²(y) + (1-sin²(x))sinh²(y)} = \sqrt{sin²(x)(cosh²(y)-sinh²(x)) + sinh²(y)} =$ [Using cosh²(y) -sinh²(y) = 1. This is a fundamental hyperbolic identity] $\sqrt{sin²(x) + sinh²(y)}$

Therefore, $|sin(x + iy)| = \sqrt{\sin^2(x) + sinh²(y)}$. For fixed x, as $|y| \to \infty, |sinh(y)| \approx \frac{1}{2}e^{|y|}$ grows exponentially, so $|\sin(x+iy)| \approx |sinh(y)| \approx \frac{1}{2}e^{|y|} \to \infty$. In particular, for purely imaginary arguments, sin(iy) = isinh(y), then $|sin(iy)| = |sinh(y)| \to \infty$ This formula clearly explains why complex sine grows exponentially in the imaginary direction.

$\sinh(y) = \frac{e^y - e^{-y}}{2}$. As $y \to +\infty$, $e^{-y}$ is negligible, so $\sinh(y) \sim \frac{1}{2} e^y$. As $y \to -\infty$, $e^{y}$ is negligible and $e^{-y}$ dominates, so $\sinh(y) \sim -\frac{1}{2} e^{-y}$, hence $|\sinh(y)| \sim \frac{1}{2} e^{|y|}$ for large $|y|$. So indeed, $\sinh(y)$ grows exponentially in magnitude as $|y| \to \infty$.

Power Functions

Let $z \in \mathbb{C}^*$ and $\alpha \in \mathbb{C}$. Complex power functions are defined using the complex logarithm: $\boxed{z^\alpha = e^{\alpha \log(z)}}$. Since the complex logarithm log(z) is multi-valued, complex powers are, in general, multi-valued functions.

Recall that $log(z) = ln|z| + i(arg(z) + 2\pi k), k \in \mathbb{Z}$. Substituting this into the previous definition: $z^\alpha = e^{\alpha \log(z)} = e^{\alpha(ln|z| + i(arg(z) + 2\pi k))} = |z|^{\alpha}e^{i\alpha (arg(z) + 2\pi k)}$. Thus, different choices of the argument lead to different values of $z^\alpha$.

Single-Valued vs Multi-Valued Behavior

The nature of $z^\alpha$ depends critically on the exponent $\alpha$.

1. Integer powers. If $\alpha \in \mathbb{Z}$, then $z^n = \underbrace{z \cdot z \cdots z}_{n\ \text{times}}$ is single-valued and independent of the choice of argument.
2. Rational powers. If $\alpha = \frac{p}{q}$ (in lowest terms), then $z^{p/q} = \left(|z| e^{i(\arg z + 2\pi k)}\right)^{p/q}$ produces exactly q distinct values. These values correspond to the q-th roots of $z^p$.

   Because the exponential function is periodic with period 2πi, two integer values $k_1$ and $k_2$ yield the same result if $\frac{p(arg(z)+2\pi k_1)}{q} = \frac{p(arg(z)+2\pi k_2)}{q} \text{ mod (2π) } \implies \frac{2\pi p(k_1-k_2)}{q} ≡ 0 \text{ mod (2π) } \implies p(k_1-k_2)$ ≡ 0 (mod q). Since p and q are coprime this is equivalent to $k_1 ≡ k_2$ (mod q). Hence, there are exactly q distinct values corresponding to k = 0, 1, …, q - 1.

3. Irrational powers. If $\alpha \notin \mathbb{Q}$, then the set $\{ e^{2\pi i k \alpha} : k \in \mathbb{Z} \}$ is infinite, yielding infinitely many distinct values of $z^\alpha$.

Principal Value of a Power Function

To obtain a single-valued branch, we define the principal power using the principal logarithm: $\boxed{z^\alpha := e^{\alpha Log(z)}}$ where $Log(z) = \ln|z| + iArg(z), Arg(z) \in (-\pi, \pi]$.

This definition yields a single-valued, continuous function on $\mathbb{C} \setminus (-\infty,0]$, with a branch cut along the negative real axis, e.g., Square Root, let $\alpha = \tfrac{1}{2}, z^{1/2} = \sqrt{z}$

Using the multi-valued logarithm, $\sqrt{z} = e^{\frac{1}{2}(\ln|z| + i(arg(z) + 2\pi k))} =[*] \pm \sqrt{|z|} e^{iarg(z)/2}.$ Thus, the square root is two-valued.

[*] Since $e^{i\pi k} = 1$ for even k and -1 for odd k, only two distinct values arise.

The principal square root is defined by $\boxed{\sqrt{z} := \sqrt{|z|} e^{iArg(z)/2}}$ with $Arg(z) \in (-\pi,\pi]$. This choice of Arg(z) places the branch cut along the negative real axis (including the origin), where $\sqrt{z}$ is discontinuous.