Exponential Function

• We define $exp(z) = e^z$ where $z$ is complex variable as below

$$ e^{z} = e^{x}e^{iy} $$

• And from the fact that $e^{iy} = cos(y) + i sin(y)$

Key Observations

• $\sqrt[n]{e^z} = e^{\frac{z}{n}}$ is set of roots of $e^z$, not a single valued function
  • $e^{z} = e^{x}e^{i(y + 2k\pi)}$ Which results in $e^{\frac{z}{n}} = e^{\frac{x}{n}}e^{i\frac{(y + 2k\pi)}{n}}$ , so for any $n<2\pi$ we have a distinct root.
• $e^{z_1+z_2} = e^{z_1}e^{z_2}$
• $e^{-z_1} = \frac{1}{e^{z_1}}$
• $\frac{d}{dz}e^z = e^z$
• $e^z$ is entire (Analytic everywhere)
• $e^z$ is periodic with a period of $2\pi{i}$
  • $e^{z+2\pi{i}} = e^{z}e^{2\pi{i}}$ and from the fact that $e^{2\pi{i}} =1$ we have $e^{z+2\pi{i}} = e^z$

Logarithms

We define $\log(z)$ as below

$$ log(z) = ln|z| + i(\Theta + 2n\pi) $$

Where

• $z$ is a complex number
• $|z|$ is the modulus (absolute value) of $z$
• $θ=arg(z)$ is the argument of $z$, defined up to an additive multiple of $2π$
• $n∈Z$ accounts for the multivalued nature of the complex logarithm.

Because of the integer parameter $n$, the expression above defines infinitely many values corresponding to different branches of the logarithm. Thus, $log(z)$ is not a function in the traditional sense (i.e., it is not single valued), but rather a multivalued function. Each choice of $n$ gives a different "branch" or value of the logarithm.