No! It is possible for a function of a complex variable to satisfy the C-R equations on a line but nowhere else. Such a function is then nowhere analytic, since it does not satisfy the C-R equations on any nonempty open set.
I have two functions u and v which are defined as harmonic within a region (they both satisfy Laplace's equation)
If I can show that u and v satisfy the Cauchy Riemann equations along a line in that region, can one show that they must satisfy the C-R everywhere?
Maybe something like a Taylors expansion of u and v along that line?
but you didn't use the fact that both u and v are harmonic at all. Harmonic functions have some sort of rigidity like an analytic function.
So the actual question is, given three equations of v:
Where u is a known harmonic function.
The first equation holds for the whole region, while the last two only hold on a line. Do the 3 equations determine v uniquely( up to a constant)?
Your argument doesn't prove or disprove this one.
I guess the answer is yes but I haven't got a proof yet.
Without loss of generality suppose the line is (part of) the x-axis.
With the two partial derivatives and known along the x-axis for a harmonic function v, we're expecting to determine v up to a constant.
Consider the function , as the partial derivative of a harmonic function, f itself is harmonic. And since we know f along the x-axis, its partial derivative is known along the x-axis, and since , and is known along the x-axis, is also known along the x-axis.
Similarly, let being harmonic, is known and is also known, along the x-axis.
Thus we got two more harmonic functions f and g, with both of their partial derivatives known along the x-axis.
Proceeding with the same steps, we get all orders of all the partial derivatives of v known along the x-axis. Since v is harmonic v is real analytic, that is, v equals to its Taylor series expansion in a neighborhood of a chosen point.
Then v is uniquely dermined along a neighborhood of the x-axis, if we choose its value at a point. Use the procedure of analytic continuation we can expand the region without crossing any singular points.