Here's the first theorem that we'll talk about today.

Suppose, f is an analytic function on a domain.

Remember what a domain is?

That is an open and connected set.

So it's all in one piece.

And suppose that you have an f prime of z,

the derivative of f, is equal to 0 everywhere in D.

The theorem says that must imply that f is constant,

which is not all that surprising because we kind of know the 1-dimensional analog.

If f is a function that is defined on an interval, and it's differentiable,

and it's derivative is 0 for all x and f is constant.

And the image that we have in mind there is, well, what else could it be?

Here's my interval from a to b.

And I'm supposed to draw a function but the derivative is 0.

Remember, the derivative to one dimension.

The derivative meant the slope of the tangent line, so

I'm supposed to draw a graph that is nice and in one piece and smooth and

it has a tangent line with a 0 derivative everywhere.

So the only way to do that is to draw something constant,

this is supposed to be constant.

So we use this one dimensional fact and use it to prove the complex fact.