What is ? What number must be if ? It can be any number.

What is ? What number must be if ? It can be any number.

Hence these expressions are undefined.

What is if ? In some cases it's . It depends on which functions and are. It can be any number or or . But it's not always undefined. In many cases it's defined and equal to a particular number. For that reason is an indeterminate form.

What is if ? Again this depends on which functions and are. In many cases it's a specific number. This is also an indeterminate form.

On the complex plane, $e^z = e^{z+2\pi ki}$ for any integer $k$. This is a bit funny, right? While the equality holds, we see that $z \ne z + 2\pi ki$ ! So if we try the inverse of the exponential function, it's a bit ambiguous as to what number we get back. Does $\log e^z = z$ or does $\log e^z = z + 2\pi i$?

The logarithm can return different complex numbers! It appears that the complex logarithm, is multi-valued. Here's a common definition:

$$\log z = \log |z| + i \arg z +i2\pi k$$

Where $\arg z$ returns the argument or angle of the complex number $z$. Note that the final term $i2\pi k$ accounts for the infinitely many values that can be returned. If we let $z=re^{i\phi}$ and substitute into the line above, we get

$$\log z = \log r + i\phi +i2\pi k$$

Then

$$\log (-1) = \log |-1| + i(2k+1)\pi = i(2k+1)\pi$$

Now this multi-valued business can be inconvenient, so we restrict the outputted values. This is called choosing a branch. A common, or principal branch is where $\phi \in [0,2\pi)$. Another common one is where $\phi \in [-\pi,\pi)$. Note that the branch we choose covers the plane entirely, and turns the multivalued function into a single valued one. Choosing the branch $\phi \in [0,2\pi)$, we can then say

$$\log_0 z = \log r + i\phi$$

where I use a subscript $0$ to denote my chosen branch. Then

$$\log_0( -1) = \log |-1| + i\pi = i\pi$$