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

Bernard Jenkins

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