Hey everyone today, I want to take a look at a problem that is initially pretty easy.
Most people familiar with popular mathematics will be able to solve it - or at least understand how solving it works.
The reason I find this problem so interesting is that it actually gives us some pretty interesting and important results when we try to generalize
it further, and while it may seem like a crazily abstract question,
it actually holds some useful applications. But now, on with the solution. If you look at the graph for natural log of x,
you'll see that not much is happening for negative values of x.
This isn't because the values don't exist, but because they're not real. To solve this problem, we need to leverage the math of complex numbers.
The trick to solving this is recalling Euler's most famous identity, e to the i pi plus 1 equals 0.
This is a beautiful equation that really deserves its own video,
but here I'm just gonna take it for granted. If we subtract 1 on both sides we end up with e to the i pi equals negative one.
Substituting this new definition for negative 1 into our expression,
We're left with natural log of e to the I pi
And by this point, we're pretty much done. Recall that the definition of the natural log is the function that cancels out exponentiation by e.
Since the natural log and e cancel each other out, we're left with a pretty satisfying answer i pi.
Now at this point, we've got a lovely-looking answer for an interesting problem,
but I'm sure many of you are wondering how we can generalize this to find the natural log of any complex number.
The math becomes a bit trickier here, but I'm sure you'll manage. In general
it's a good idea to break up this set into a different form. One go to is the exponential polar form.
z equals r e to the i theta?
Hopefully you'll see that having that e there will come in handy shortly. For our definition of the complex logarithm,
we'll need it to satisfy the same properties as a real function natural log of x,
so we'll want it to follow the log sum rule that says natural log of a times b
equals natural log of a plus natural log of b.
So, from that, we can see that natural log of r times e to the i theta
must equal natural log of r plus natural log of e to the i theta. Once again,
we can see that the natural log and e cancel each other out here. From that we get natural log of r e to the
i theta equals natural log of r plus i theta.
Hopefully you'll be able to see that since r is a positive real number,
the natural log of r will also be real. And since theta is a real number, i theta is
strictly imaginary. So now we've broken natural log of Z into its real and imaginary components. One interesting thing of note
is that by taking the logarithm of a complex number its real part is only dependent on its length and its imaginary part is only
dependent on its angle. In the description
I've linked a video by Matt Parker that uses this fact to create a droste effect for spherical videos - definitely worth the watch.
By this point most of you will be pretty happy with what we've done so far, but there is a slight caveat.
What if we add a multiple of 2 pi to theta?
Geometrically, adding 2 pi to any angle is like multiplying by 1 - you end up in the same place.
If we go back to how we define z,
we let it to be some number our e to the i theta.
But what if we want to solve the general problem? We need to let z equal r e to the (theta plus 2 pi k) i
where k is some integer. If we do the same algebra as before,
we're left with natural log of z equals natural log of r plus (theta plus 2 pi k) i.
This isn't good since the same number could get us a different natural log
depending on what we set k equal to. To fix this we usually
restrict theta to be between negative pi and pi. We call this the principal log, and denote it with a capital L, Log(z).
Finding the complex definition of real functions ends up being a pretty useful idea in all sorts of situations,
so I implore you to find out how we define cosine and sine over the complex numbers.
It requires a pretty clever trick using the general version of Euler's formula.
Although it's an interesting trick. There's not much content there,
so I probably won't make a video about it unless I end up doing a series on complex numbers. In this video,
we learned how to take the natural log of complex numbers. If you have any questions or spotted any mistakes,
feel free to leave a comment.