Exponential Function for Complex Numbers

Calculation of \(e^z\) using Euler's formula

Exp Function Calculator

Exponential Function \(e^z\)

The exponential function for complex numbers is calculated by Euler's formula: \(e^{x+iy} = e^x(\cos y + i\sin y)\). It is the inverse of the complex logarithm.

Complex number z = x + iy (exponent)
+
i
Calculation Result
\(e^z\) =

Exp Properties

Euler's Formula
\[e^{iy} = \cos(y) + i\sin(y)\]

One of the most beautiful formulas in mathematics!

Complete Formula
\[e^{x+iy} = e^x(\cos y + i\sin y)\]
Magnitude \(|e^z| = e^x\)
Argument \(\arg(e^z) = y\)
Important Properties
  • \(e^{z_1 + z_2} = e^{z_1} \cdot e^{z_2}\) (Functional equation)
  • \(e^z \neq 0\) for all \(z \in \mathbb{C}\)
  • \(e^{z + 2\pi i} = e^z\) (Periodicity)
  • Inverse: \(\ln(e^z) = z + 2\pi i k\)
Special Cases
  • \(e^0 = 1\)
  • \(e^{i\pi} = -1\) (Euler's identity)
  • \(e^{i\pi/2} = i\)
  • \(e^{2\pi i} = 1\)


Formula for the Exponential Function of Complex Numbers

The exponential function for complex numbers extends the real exponential function by using Euler's formula.

Definition

For \(z = x + iy\):
\[e^z = e^{x+iy} = e^x \cdot e^{iy}\] \[= e^x(\cos y + i\sin y)\]

Components

Real part: \(\text{Re}(e^z) = e^x \cos y\)
Imaginary part: \(\text{Im}(e^z) = e^x \sin y\)

Calculation Example

Calculation: \(e^{3+5i}\)
Step 1: Components

Given: \(z = 3 + 5i\)

Real part: \(x = 3\)

Imaginary part: \(y = 5\)

Step 2: Euler's formula

\(e^{3+5i} = e^3 \cdot e^{5i}\)

\(= e^3(\cos 5 + i\sin 5)\)

Step 3: Exponential

\(e^3 \approx 20.086\)

Scale factor for real and imaginary parts

Step 4: Trigonometric values

\(\cos(5) \approx 0.284\)

\(\sin(5) \approx -0.959\)

(Angles in radians!)

Step 5: Final result

Real part: \(20.086 \cdot 0.284 \approx 5.70\)

Imaginary part: \(20.086 \cdot (-0.959) \approx -19.26\)

\(e^{3+5i} \approx 5.70 - 19.26i\)

Geometric Interpretation

Magnitude: \(|e^{3+5i}| = e^3 \approx 20.086\)
The magnitude depends only on the real part!

Argument: \(\arg(e^{3+5i}) = 5\) rad
The argument is the imaginary part!

Important Properties and Theorems

Functional equation
\[e^{z_1 + z_2} = e^{z_1} \cdot e^{z_2}\]

The exponential function turns addition into multiplication.

Periodicity
\[e^{z + 2\pi i k} = e^z\]

The complex exponential function is periodic with period \(2\pi i\).

No zeros
\[e^z \neq 0 \quad \forall z \in \mathbb{C}\]

The exponential function has no zeros in the complex plane.

Euler's identity
\[e^{i\pi} + 1 = 0\]

The most famous formula in mathematics connects the five most important constants: \(e\), \(i\), \(\pi\), 1 and 0.

Polar form relation
\[z = r e^{i\phi} = r(\cos\phi + i\sin\phi)\]

Any complex number can be represented in polar form with the exponential function: \(r = |z|\), \(\phi = \arg(z)\).

Derivative
\[\frac{d}{dz}e^z = e^z\]

The exponential function is its own derivative - also in the complex domain!

Exponential Function - Detailed Description

Leonhard Euler

The Euler's formula was discovered by the Swiss mathematician Leonhard Euler (1707-1783) and is considered one of the most beautiful formulas in mathematics.

Derivation via series expansion:
\[e^{ix} = \sum_{n=0}^{\infty}\frac{(ix)^n}{n!}\] \[= 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + \ldots\] \[= \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots\right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\right)\] \[= \cos(x) + i\sin(x)\]

Practical Applications

  • Electrical engineering: alternating current calculations, impedance
  • Signal processing: Fourier transform
  • Quantum mechanics: wave functions
  • Vibration theory: complex amplitude

Geometric Meaning

Multiplication by \(e^{i\phi}\) corresponds to a rotation by angle \(\phi\) in the complex plane.

Rotation of a complex number:
\[z' = z \cdot e^{i\phi}\] This rotates the number \(z\) by angle \(\phi\) counterclockwise about the origin.

Relation to the Logarithm

The exponential function is the inverse function of the complex logarithm:

\[w = e^z \quad \Leftrightarrow \quad z = \ln(w)\]

Note: The complex logarithm is multivalued!
\(\ln(e^z) = z + 2\pi i k\) with \(k \in \mathbb{Z}\)

Important Notes

Be careful with angles!
  • Angle \(y\) in Euler's formula must always be in radians
  • Conversion: \(1° = \frac{\pi}{180}\) rad
  • Observe periodicity: \(e^{iy}\) has period \(2\pi\)
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