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Imaginary Numbers Are Real [Part 12: Riemann's Solution]

last time we left off wondering why some paths on W plane
led us to completely new values on our Z plane, while others didn't
Gauss's student Bernhard Riemann made some powerful insights into problems like
this in the mid-nineteenth century. The first part of Riemann contribution
is the idea that for problems like this
we need more than two complex planes to visualize our function. Since each point
on W plane map to two points on our Z plane we can begin to resolve our
ambiguity by adding a second W plane and letting each of our two points on Z
map to its very own copy of the W plane.
So, that's fine, but it immediately raises an important question: How do we pick
which Z values to map to each plane? A simple and effective approach here is to
simply divide the Z plane into two halves. We will let the right half map to
our first W plane and the left half map to our second W plane.
These restricted versions of our multifunction are called ★branches★.
Let's draw a path again, but this time just on our first W plane. Things look
just fine until we cross the negative real axis and our path on the Z plane
suddenly jumps!
This of course is what must happen we have required points from our first W
plane to only map to the right side of our Z plane. Almost every point on W has
two possible solutions on Z
and with our first branch we've decided to always pick the one on the right,
so our path now jumps around the Z plane, but what's perhaps more disappointing
here is that we haven't gained any insight into our interesting loop
behavior we saw last time.
In fact we can even recreate the set up, no matter what kind of loopy draw we
always end up exactly where we started on both the Z and W planes.
It seems we have legalized this behavior out of existence.
Further, the fact that our function jumps across the Z plane means that our
branches are discontinuous, a huge problem mathematically. Functions of
complex variables are a big part of modern mathematics and science and if
our functions are jumping around like this
we can't do important things, like take derivatives and integrals.
So we fix the multivalued problem by splitting our multi function into
branches, our function is now one to one.
But in the process we have introduced some serious issues. Thus far Riemann
solution is not looking so great.
Fortunately that was just part 1 and part 2 is much cooler (!)
Let's consider our discontinuity problem in a bit more detail will switch back to
our forward function momentarily, and again drawn our Z plane.
Let's pay careful attention to where just continuity show up we'll follow the
points along a single path and to make sure we can tell our points apart, we will
continuously change the color of our path.
As we move from quadrant 1 to quadrant 2 on Z we switch branches, we switch back
to our first branch when moving from quarter three to four, for our path to be
continuous
we need to somehow connect the two W planes at the exact point to our path jumps.
What Riemann saw here was a way to bring together are too complex planes in such
a way that are multi function would be perfectly continuous while maintaining
the nice one to one properties of two W plane solution.
Our next step is to grab some scissors and tape we'll figure out what to do
with them next time.
ok
yeah