Inverse

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too many directions now

We know that every nonzero Complex number can be written as the product of a positive real number (a scalar) and a Complex number on the unit circle. Euler’s Formula tells us how each Complex number on the unit circle can be written as a Complex exponential.

$\blacktriangleright $ Euler’s Formula

Every complex number on the unit circle can be written in the form

\[ e^{i \theta } = \cos (\theta ) + i \sin (\theta ) \]

And, we already know how to write any real number in exponential form: $r = e^{\ln (r)}$.

$\blacktriangleright $ Complex Numbers

Combining these together, we get that every nonzero complex number can be written in the form

\[ r \cdot (\cos (\theta ) + i \sin (\theta )) = e^{\ln (r)} \cdot e^{i \, \theta } = e^{\ln (r) + i \, \theta } \]

If $z = a + b \, i$, then $r = \sqrt {a^2 + b^2}$ and $\theta $ is the counterclockwise angle from the positive real-axis to $z$.

EVERY complex number can be written in the form $e^{\ln (r) + i \, \theta }$

Complex Logarithm

Let $z = -3 - 3 \, i$.

$|z| = \sqrt {(-3)^2 + (-3)^2} = \sqrt {18} = 3 \sqrt {2}$

$\arctan \left (\frac {-3}{-3}\right ) = \frac {\pi }{4}$. This reference angle gives us $\theta = \frac {5 \pi }{4}$, since $z$ is in the third quadrant.

$z = -3 - 3 \, i = e^{\ln (3 \sqrt {2}) + \left ( \tfrac {5 \pi }{4} \pm 2 k \pi \right ) \, i} \, \text { where } \, k \in \mathbb {N} $

There are many complex logarithms for $-3 - 3 \, i$. One for each turn around the unit circle.

There are an infinite number of complex logarithms for each complex number.

Whoops: Just like when we were creating the inverse trigonometric functions, this Complex logarithm is not following our one and only rule for functions.

We will fix this in the same waythat we fixed $\arcsin $ and $\arccos $. We’ll pick a range.

Following the idea behind our range restrictions for $\arcsin $ and $\arccos $, we pick a principal value for the complex logarithm. Since we would like to keep the positive real axis as clean as possible, we chooose the interval $(-\pi , \pi ]$ for the principal value of the complex logarithm.

This interval is known as the principal branch of the complex logarithm. The negative real axis is called a branch cut for the complex logarithm. It is a cut, because the angle changes abruptly from $\pi $ to $-\pi $.

Complex Logarithm

If $z = a + b \, i$, then

\[ \ln (z) = \ln (\sqrt {a^2 + b^2}) + \theta \, i \]

where $\theta \in (-\pi , \pi ]$ and is the counterclockwise angle from the positive real axis to $z$.

This $\theta $ is often referred to as the argument of $z$, $Arg(z)$.

Since $Arg(r) = 0$ for any positive real number, our complex logarithm will agree with our real logarithm.

Some Logarithms

$\ln (5) = \ln (5) + 0 \, i = \ln (5)$
$\ln (2i) = \ln (2) + \frac {\pi }{2} \, i$
$\ln (-5 + 5 \, i) = \ln (5 \sqrt {2}) + \frac {3\pi }{4}\, i$
$\ln (-1) = \pi \, i$

We are bringing over all of our exponent arithmetic.

$i^i$

\[ i^i = e^{\ln (i^i)} = e^{i \cdot \ln (i)} = e^{ i \cdot \tfrac {\pi }{2} \, i} = e^{-\tfrac {\pi }{2}} \approx 0.2078795764\]

$i^i$ is a real number!

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more examples can be found by following this link
More Examples of Complex Logarithms

2025-05-17 22:35:51