# Thread: The Cauchy Riemann equations and harmonic functions

1. ## The Cauchy Riemann equations and harmonic functions

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?

2. 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.

3. Originally Posted by Bruno J.
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.
That does not prove or disprove the original statement.

4. Originally Posted by xxp9
That does not prove or disprove the original statement.
There is no statement in the original post, unless I do not understand what a statement is.

The post was a question, which I answered.

5. 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:
$\Delta v = 0, \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
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.

6. deleted.

7. Without loss of generality suppose the line is (part of) the x-axis.
With the two partial derivatives $v_x$ and $v_y$ known along the x-axis for a harmonic function v, we're expecting to determine v up to a constant.
Consider the function $f=v_x$, as the partial derivative of a harmonic function, f itself is harmonic. And since we know f along the x-axis, its partial derivative $f_x$ is known along the x-axis, and since $f_y=v_{xy}=v_{yx}$, and $v_y$ is known along the x-axis, $f_y$ is also known along the x-axis.
Similarly, let $g=v_y$ being harmonic, $g_x$ is known and $g_y=v_{yy}=-v_{xx}=-f_x$ 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.