Arc Tangent (arctan) for Complex Numbers

Calculation of arctan(z) - the inverse function of tangent

Arctan Calculator

Arc Tangent (arctan)

The arc tangent arctan(z) is the inverse function of tangent: If \(\tan(w) = z\), then \(w = \arctan(z)\). For complex numbers, the function is multivalued and has infinitely many values.

Tangent value z = a + bi
+
i
Calculation Result
arctan(z) (principal value) =
The function is multivalued: All values are \(w + \pi k\) with \(k \in \mathbb{Z}\)

Arctan - Properties

Formula
\[\arctan(z) = \frac{i}{2}\ln\left(\frac{i+z}{i-z}\right)\]

With complex logarithm

Definition
\[\tan(\arctan(z)) = z\]
Real numbers ℝ → (-π/2, π/2)
Complex Multivalued
Important Properties
  • Inverse function of tan(z)
  • Multivalued: \(w + \pi k, k \in \mathbb{Z}\)
  • Principal value: \(\text{Re}(w) \in (-\pi/2, \pi/2)\)
  • \(\arctan(-z) = -\arctan(z)\) (odd function)
Relations
  • \(\arctan(z) = \frac{1}{2i}\ln\left(\frac{1+iz}{1-iz}\right)\)
  • \(\tan(\arctan(z)) = z\) (definition)
  • \(\arctan(\tan(z)) = z + \pi k\)
  • \(\arctan(1/z) = \frac{\pi}{2} - \arctan(z)\) (for z>0)


Formulas for Arc Tangent of Complex Numbers

The arc tangent arctan(z) is the inverse function of tangent and is defined by the complex logarithm.

Main Formula
\[\arctan(z) = \frac{i}{2}\ln\left(\frac{i+z}{i-z}\right)\]

With \(\ln\) = complex logarithm

Alternative Form
\[\arctan(z) = \frac{1}{2i}\ln\left(\frac{1+iz}{1-iz}\right)\]

Equivalent representation

Arc Tangent - Detailed Description

Definition and Meaning

The arc tangent (also arctan or atan) is the inverse function of the tangent function.

Definition:
\[\tan(\arctan(z)) = z\]
The arc tangent returns the angle (in radians)
whose tangent has the value z.

Notation:
arctan(z), atan(z), or \(\tan^{-1}(z)\)

For Real Numbers

For real numbers \(x \in \mathbb{R}\):

Range:

\[\arctan(x) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\]

• arctan(∞) = π/2 ≈ 1.5708 rad = 90°
• arctan(0) = 0
• arctan(-∞) = -π/2 ≈ -1.5708 rad = -90°

For Complex Numbers

For complex numbers, arctan is multivalued:

Multivaluedness:
If \(w = \arctan(z)\), then
\[w + \pi k \quad (k \in \mathbb{Z})\]
are also valid solutions.

Principal value:
The principal value has \(\text{Re}(w) \in (-\pi/2, \pi/2)\)

Periodicity:
Period π (not 2π like sin/cos!)

Important Relations

  • \(\arctan(-z) = -\arctan(z)\) (odd function)
  • \(\arctan(1/z) = \frac{\pi}{2} - \arctan(z)\) (for z>0)
  • \(\arctan(\overline{z}) = \overline{\arctan(z)}\)
  • \(\arctan(z) = \arcsin\left(\frac{z}{\sqrt{1+z^2}}\right)\)

Singularities

Caution at z = ±i:
arctan(i) and arctan(-i) are undefined!
These are poles of the function.
The tangent is not injective at these values.

Geometric Meaning (Real Numbers)

Right Triangle:
In a right triangle:
\[\tan(\alpha) = \frac{\text{Opposite}}{\text{Adjacent}}\]
The arc tangent calculates the angle α:
\[\alpha = \arctan\left(\frac{\text{Opposite}}{\text{Adjacent}}\right)\]
Example:
Opposite: a = 4
Adjacent: b = 3
\[\tan(\alpha) = \frac{4}{3} \approx 1.333\]
\[\alpha = \arctan(1.333) \approx 0.9273 \text{ rad}\]
\[\alpha \approx 53.13°\]

atan2 Function

Two-Argument Arc Tangent

The atan2(y, x) function calculates \(\arctan(y/x)\) and automatically considers all four quadrants:

\[\text{atan2}(y, x) = \arctan\left(\frac{y}{x}\right) + \text{quadrant correction}\]

Range: (-π, π] (full circle!)
Advantage: Avoids division by zero and provides correct quadrant

Calculation Examples

Example 1: arctan(1)

Real number: z = 1

\(\arctan(1) = \frac{\pi}{4}\)

≈ 0.7854 rad = 45°

Example 2: arctan(√3)

z = √3 ≈ 1.732

\(\arctan(\sqrt{3}) = \frac{\pi}{3}\)

≈ 1.0472 rad = 60°

Example 3: arctan(0)

Zero point: z = 0

\(\arctan(0) = 0\)

= 0 rad = 0°

Example 4: arctan(0.4 + 0.3i)

Complex number: z = 0.4 + 0.3i

Use formula:

\(\arctan(z) = \frac{i}{2}\ln\left(\frac{i+z}{i-z}\right)\)

Result: see calculator above

Example 5: arctan(i) - Singularity!

z = i (imaginary unit)

\(\arctan(i)\) is undefined!

Pole of the function

Example 6: arctan(2i)

Purely imaginary: z = 2i

\(\arctan(2i) = \frac{i}{2}\ln\left(\frac{i+2i}{i-2i}\right)\)

≈ 0 + 0.549i

Special Values (real)
arctan(0)
= 0
arctan(1)
= π/4 = 45°
arctan(√3)
= π/3 = 60°
arctan(∞)
→ π/2 = 90°

Applications

Geometry & Navigation
  • Angle calculation from side ratio
  • Determining slope angle
  • Direction angle (atan2)
  • Polar coordinates
Physics & Engineering
  • Projectile trajectory (launch angle)
  • Impedance in AC circuits
  • Phase shift
  • Signal processing
Mathematics
  • Complex analysis
  • Integral calculus
  • Differential equations
  • Numerical methods
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