Natural Logarithm of Complex Numbers
Calculation of \(\ln(z)\) - the inverse of the complex exponential function
Logarithm Calculator
Natural Logarithm \(\ln(z)\)
The natural logarithm (base \(e\)) of a complex number is the inverse of the exponential function. The logarithm is multivalued and here the principal value is returned.
Logarithm - Properties
Principal Value
With \(-\pi < \arg(z) \leq \pi\)
Components
Imaginary part: \(\text{Im}(\ln z) = \arg(z) = \arctan\left(\frac{b}{a}\right)\)
Multivaluedness
In general: \(\ln(z) = \ln|z| + i(\arg(z) + 2\pi k)\) with \(k \in \mathbb{Z}\)
Infinitely many values!
This calculator returns the principal value (k=0)
Important Properties
- \(\ln(z_1 \cdot z_2) = \ln(z_1) + \ln(z_2)\) (modulo \(2\pi i\))
- \(\ln(z_1 / z_2) = \ln(z_1) - \ln(z_2)\) (modulo \(2\pi i\))
- \(\ln(z^n) = n\ln(z)\) (modulo \(2\pi i\))
- \(e^{\ln(z)} = z\) (unique)
Inverse Function
\(\ln(z)\) is the inverse of \(e^z\):
If \(w = e^z\), then \(z = \ln(w)\)
Formulas for the Natural Logarithm
The natural logarithm of a complex number \(z = a + bi\) is computed by:
Standard formula
With \(|z| = \sqrt{a^2+b^2}\) and \(\arg(z) = \arctan(b/a)\)
Component form
Direct computation from real and imaginary parts
Calculation Example
Calculation: \(\ln(3 + 5i)\)
Step 1: Given
\(z = 3 + 5i\)
Real part: \(a = 3\)
Imaginary part: \(b = 5\)
Step 2: Real part
\(\text{Re}(\ln z) = \frac{1}{2}\ln(a^2+b^2)\)
\(= \frac{1}{2}\ln(3^2 + 5^2)\)
\(= \frac{1}{2}\ln(9 + 25)\)
\(= \frac{1}{2}\ln(34)\)
\(\approx 1.763\)
Step 3: Imaginary part
\(\text{Im}(\ln z) = \arctan\left(\frac{b}{a}\right)\)
\(= \arctan\left(\frac{5}{3}\right)\)
\(\approx 1.030\) rad
(≈ 59.04°)
Step 4: Result
\[\ln(3 + 5i) = 1.763 + 1.030i\]
The imaginary part is given in radians
Verification
Check with exponential:
\(e^{1.763+1.030i} = e^{1.763} \cdot e^{1.030i}\)
\(= 5.831 \cdot (\cos 1.030 + i\sin 1.030)\)
\(= 5.831 \cdot (0.515 + 0.857i)\)
\(\approx 3.0 + 5.0i\) ✓
Alternative computation:
\(|z| = \sqrt{3^2+5^2} = \sqrt{34} \approx 5.831\)
\(\ln|z| = \ln(5.831) \approx 1.763\) ✓
\(\arg(z) = \arctan(5/3) \approx 1.030\) ✓
Multivalued Nature of the Complex Logarithm
Problem: Infinitely many values
The complex logarithm is not unique!
with \(k \in \mathbb{Z}\) (any integer)
Reason: The exponential function is periodic: \(e^{z+2\pi i} = e^z\)
Example: \(\ln(1)\)
Possible values:
- \(k=0\): \(\ln(1) = 0\) (principal value)
- \(k=1\): \(\ln(1) = 2\pi i\)
- \(k=-1\): \(\ln(1) = -2\pi i\)
- \(k=2\): \(\ln(1) = 4\pi i\)
- etc. (infinitely many!)
Solution: Principal Value
To obtain uniqueness, define the principal value:
with \(-\pi < \arg(z) \leq \pi\) (principal branch)
This corresponds to k=0 in the general formula
Branch Cut
The principal value has a branch cut along the
negative real axis. When crossing it the imaginary part jumps by \(2\pi\).
Note: For negative real numbers the imaginary part is \(\pm\pi\)
(depending on convention)
Graphical representation of multivaluedness
Principal: \(0\)
Others: \(2\pi k i\)
Principal: \(\pi i\)
Others: \(\pi i + 2\pi k i\)
Principal: \(\frac{\pi}{2}i\)
Others: \(\frac{\pi}{2}i + 2\pi k i\)
Natural Logarithm - Detailed Description
Inverse of the Exponential Function
The natural logarithm \(\ln(z)\) is the inverse of the complex exponential function \(e^z\).
• If \(w = e^z\), then \(z = \ln(w)\)
• It holds: \(e^{\ln(z)} = z\) (unique)
• But: \(\ln(e^z) = z + 2\pi ik\) (multivalued!)
• Principal value: \(\ln(e^z) = z\) for \(\text{Im}(z) \in (-\pi, \pi]\)
Computation using Polar Form
If \(z = r e^{i\phi}\) in polar form, then:
With \(r = |z|\) and \(\phi = \arg(z)\)
Practical Applications
The complex logarithm finds applications in many areas:
• Complex analysis: conformal mappings
• Electrical engineering: Bode diagrams, filter design
• Signal processing: spectral analysis
• Physics: quantum mechanics, fluid dynamics
Computation rules
Be careful with computation rules!
The familiar logarithm rules only hold modulo \(2\pi i\):
• \(\ln(z_1 \cdot z_2) = \ln(z_1) + \ln(z_2) + 2\pi ik\)
• \(\ln(z^n) = n\ln(z) + 2\pi ik\)
• The principal value satisfies the rules only approximately!
Special cases
- Positive real numbers: \(\ln(a) = \ln(a) + 0i\) (as real)
- Negative real numbers: \(\ln(-a) = \ln(a) + \pi i\)
- Imaginary unit: \(\ln(i) = \frac{\pi}{2}i\)
- One: \(\ln(1) = 0\) (principal value)
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More complex functions
Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •Cosh • Sinh • Tanh •
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Airy function • Derivative Airy function •
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