Holomorphic function?

Question

Ok, this has been bugging for some time now. Can someone please explain to me what the heck is the difference between a function being holomorphic and a function being differentiable?

They look like the SAME EXACT THING but my textbook says that being holomorphic is stronger than being differentiable.

Answer 1

This is a great question, and you make a great observation. The difference between the two is subtle but is very important.

It's true that the FORM of the limit looks the same. However, a "differentiable" function's limit at a point only has to exist when approaching the point from JUST the REAL LINE. That is, you approach the limit from the "left" and the "right." A "holomorphic" function's limit at a point has to exist when approaching the point from EVERY DIRECTION in the COMPLEX PLANE. This is like approaching the point from every direction on a DISC rather than just on a line.

If you look very closely in your book, I think you'll see that the holomorphic limit in the book uses a point z from the complex plane C whereas the differentiable limit in the book uses a point x from the real line R.

For example, if I wanted to take the limit as something is approaching 0, I might want to look at the function's behavior at +delta and -delta for some small delta > 0. However, I COULD look at the function's behavior at a -i*delta and i*delta. Similarly, I could look at the function's behavior at delta*exp(i*theta), where theta is any angle. The differential limit only cares about the real cases. The holomorphic limit cares about all the complex cases too.

It should be pretty clear that if a function is holomorphic, it is differentiable. (actually, a holomorphic function is infinitely differentiable! In other words, begin holomorphic is MUCH stronger than being once differentiable!!) In fact, holomorphic functions are sometimes called "complex-differentiable."

NOTE: It is not enough to say that satisfying the Cauchy-Riemann equations is equivalent to being holomorphic. To quote the "Holomorphic" source I list below, "A complex function f(x + iy) = u + iv is holomorphic if and only if it satisfies the Cauchy-Riemann equations and u and v have continuous first partial derivatives with respect to x and y." Because the Cauchy-Riemann equations are necessary BUT NOT SUFFICIENT, then they alone do not encapsulate the true meaning of being holomorphic. When you think holomorphic, you should think "complex differentiable." You should consider the typical limit that you would use to define differentiability, but apply the limit as you approach from all directions in the complex plane. This is a mature understanding of "holomorphic" that is very useful when studying analytic functions (even in REAL analysis).

Answer 2

a holomorphic function
satisfies an equation (Cauchy Riemann equation), a differentiable function does not necessarily satisfies this equation

an example would be good:
f(z)=conjugate of z
this function is differentiable, but it is not holomorphic. I am sure you have the definitions in your book, but this also has de definition:
http://mathworld.wolfram.com/Cauchy-RiemannEquations.html

so, if z=x+iy f(x,y)=u(x,y)+iv(x,y),
now in the example above f(x,y)=x-iy, so u(x,y)=x, and v(x,y) = -y
now,
partial of u with respect to x = 1
and the partial of v with respect to y is -1,
and therefore are not equal, so the first Cauchy-Riemann equation is not satisfied.
nevertheless, as functions of x and y, u and v, and therefore f,
are differentiable functions, since you can take their derivatives as many times as you wish

hint: the answers below are all nonsense....

Answer 3

Are you sure of the spelling? Homomorphic, isomorphic and endomorphic I seem to remember. But not holomorphic. However, I could be out of touch now.

Answer 4

holomorphic means complex-differentiable.
differentiable is used for real valued functions.
the difference is that comple-differentiable means that the limit exists no matter what direction you approach to some point.
Differentiable you have the left and right side limit.