What is the Difference Between Holomorphic and Analytic Functions?

[Jex]

Okay I have that a function is analytic at point a if it can be represented by a power series in x-a with a positive radius of convergence.

Does anyone else have a better way of putting this?

Thanks

 

[Triss]

Let $$\Omega\subseteq\mathbb{C}$$ be an open connected set. Then $$f:\Omega\to\mathbb{C}$$ is analytic in $$\Omega$$, if and only if $$f$$ is holomorphic in $$\Omega$$.

 

[HallsofIvy]

Iriss, what does it mean to say a function is "holomorf" ("holomorphic"?) in a set. My understanding of "holomorphic" is that it is a synonym for "analytic" so you haven't really answered the question: Was what he gave a good characterization of "analytic" or "holomorphic"?

Jex, the only change I would make in your definition would be to say
"is equal to" rather than "can be represented by" since "represented by" is vague.
The function
[tex]f(x)= e^{-\frac{1}{x^2}} if x\ne 0, 0 if x= 0[/itex]
has a power expansion (Taylor's series) about 0 that converges for all x. Is it "represented" by that series?
Of course, the series converges for all x because it is identically 0! f(x) is not equal to the series anywhere except x= 0.

 

[Jex]

Thanks guys.

 

[matt grime]

There are other equivalent definitions, such as the Cauchy Riemann equations, Morera's theorem, and the fact that the real and imaginary parts should be harmonic (which is just a restatement of C-R equations), as well as the fact that it is equivalent to complex differentiability.

 

[Triss]

Iriss, what does it mean to say a function is "holomorf" ("holomorphic"?) in a set. My understanding of "holomorphic" is that it is a synonym for "analytic" so you haven't really answered the question: Was what he gave a good characterization of "analytic" or "holomorphic"?

A function is holomorphic in a set if it is holomorphic in every point in the set. The point is that not all holomorphic functions are analytical in any given subset of $$\mathbb{C}$$.

 

[shmoe]

The point is that not all holomorphic functions are analytical in any given subset of $$\mathbb{C}$$.

I don't understand your point, 'tis obvious that the terms "holomorphic" and "analytic" apply to a function on some subset of points, no? Or are you making some distinction between "analytic" and "holomorphic"? They are synonyms in every usage I've ever encountered ("regular" is another synonym).

 

[Triss]

I don't understand your point, 'tis obvious that the terms "holomorphic" and "analytic" apply to a function on some subset of points, no? Or are you making some distinction between "analytic" and "holomorphic"? They are synonyms in every usage I've ever encountered ("regular" is another synonym).

I am making a distinction between "analytic" and "holomorphic". Oviously they are they same when in the complex plane, but the same does not apply on $$\mathbb{R}$$. Here there are holomorphic functions that are not analytic, the function in HallsofIvy's post being a good example.

 

[shmoe]

What definition are you using that considers Hall's example to be "holomorphic" in any way? It's not one I've ever seen before.

 

[Triss]

What definition are you using that considers Hall's example to be "holomorphic" in any way? It's not one I've ever seen before.

Let $$\Omega\subseteq\mathbb{C}$$ be an open set and $$f:\Omega\to\mathbb{C}$$ a function. Then $$f$$ is holomorphic at a point $$z_0\in\Omega$$ if for all $$z\in\mathbb{C}$$ the limit

$$\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$$

exists. In the case of
$$f(x)= \begin{cases} \exp{(-1/x^2)} & x\neq 0\\ 0 & x=0 \end{cases}$$
since this is defined on $$\mathbb{R}$$, the limit

$$\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}$$

exists for all $$x\in\mathbb{R}$$ and the function is holomorphic/differentiable.

 

[shmoe]

Your definition of a holomorphic function on the reals is the same as it being differentiable (in the real sense) once. I find that an odd thing to do and would like to know what source you are getting this from (if any?).

 

[Triss]

Your definition of a holomorphic function on the reals is the same as it being differentiable (in the real sense) once. I find that an odd thing to do and would like to know what source you are getting this from (if any?).

Good point. You would like the function to should be differentiable infinitively many times?

 

[shmoe]

Good point. You would like the function to should be differentiable infinitively many times?

I would like to know where your definition of holomorphic applied to real functions came from.

The concept of analytic/holomorphic/regular for complex valued functions just requires one complex derivative to get derivatives of all orders and equality to a power series. The definition of "real analytic" functions captures this consequence-derivatives of all orders and equal to power series as one real derivative is not enough. I can't say I've seen anyone bother defining "real holomorphic" functions, but if they did I can't possibly see why it would differ from "real analytic".

 

[Triss]

Well then it is just a question of what we call it. Being picky the definition of a function being holomorphic only apply to complex valued functions. So defining "real holomorphic functions" could be something I just did in the spur of the moment. So the reason I made the distinction between holomorphic and analytic earlier was because in the real case there are (infinitely) differentiable functions that is not analytic.

 

[matt grime]

So defining "real holomorphic functions" could be something I just did in the spur of the moment.

Ok, it could be, but was it? And if so why did you do it?

 

[shmoe]

So defining "real holomorphic functions" could be something I just did in the spur of the moment.

This is an exceptionally bad thing to do, unless you explain that this isn't standard terminology and you're just making it up as you go along.

So the reason I made the distinction between holomorphic and analytic earlier was because in the real case there are (infinitely) differentiable functions that is not analytic.

This doesn't explain why you would want to distinguish between "holomorphic" and "analytic", they are synonyms.

 

[HallsofIvy]

I have seen (don't remember where right now) a function from R to R being defined to be "real analytic" on a set if and only if it is equal to it's Taylor's series on that set, (Which of course requires that it be infinitely differentiable in order that the series exist. The point of my example before was that a function may be infinitely differentiable (on the real numbers) without being "analytic".)

I am making a distinction between "analytic" and "holomorphic". Oviously they are they same when in the complex plane, but the same does not apply on . Here there are holomorphic functions that are not analytic, the function in HallsofIvy's post being a good example.

Iriss, you seem to be saying that "homorphic" applies to a set while "analytic" necessarily applies to all of C. That's not any notation I have ever seen. A function is "entire" if it is analytic on all of C.

 

[Triss]

Ok, it could be, but was it? And if so why did you do it?

Atleast I have not found any refrence to support it. Since a function is holomorphic if it is has a complex derivative it was easy to cut it down to also mean the same for real valued functions. Even more so because of that nice little function that is infinitely differentiable but not analytic. Though as shmoe reprimand, that is not the idea idea I have had.

This doesn't explain why you would want to distinguish between "holomorphic" and "analytic", they are synonyms.

I think it does. We have just 'agreed' that i cannot use the term holomorphic for real functions, so the point of making a diffrence between "holomorphic" and "analytic" is not an issuse anymore.

Triss, you seem to be saying that "homorphic" applies to a set while "analytic" necessarily applies to all of C. That's not any notation I have ever seen. A function is "entire" if it is analytic on all of C.

I hope I am not saying that :)

 

[shmoe]

Since a function is holomorphic if it is has a complex derivative it was easy to cut it down to also mean the same for real valued functions.

This is also a definition for analytic complex functions. So:

I think it does. We have just 'agreed' that i cannot use the term holomorphic for real functions, so the point of making a diffrence between "holomorphic" and "analytic" is not an issuse anymore.

I would still say this is no explanation for considering them different. Your attempt to apply holomorphic to real functions essentially just changed 'complex' to 'real', yet you already agree to do something different when it comes to the term 'analytic'. This seems like one of the most arbitrarily confusing way of doing things, and serves no purpose whatsoever that I can see.

 

[Jex]

I know you might be curious about "holomorphic" and what not but don't you think we are taking this beside the point? Some of us reading this are not that advanced in mathematics to even understand what some of you are saying. Which is not a bad thing at all, I will get there eventually but right now I am nearing the end of my diff eq class and really the discussion going on here is just confusing me worse on my original question. I didn't know my question that I thought was pretty simple was going to strike a mini debate on terms. So to go back to my original post, what do some of you have to say to the following on analycity:

"In calculus it is seen that functions such as cosx and ln(x-1) can be represented by power series by expansions in either Maclaurin or Taylor series. It is said that these functions are analytic."

So basically this is my original question. Is this a strong definition (a reasonable definition for someone who has only taken up to diff eq)? Without getting wrapped up in complex numbers, whether or not it is synonymous with holomorphic, etc.

 

[HallsofIvy]

I know you might be curious about "holomorphic" and what not but don't you think we are taking this beside the point? Some of us reading this are not that advanced in mathematics to even understand what some of you are saying. Which is not a bad thing at all, I will get there eventually but right now I am nearing the end of my diff eq class and really the discussion going on here is just confusing me worse on my original question. I didn't know my question that I thought was pretty simple was going to strike a mini debate on terms. So to go back to my original post, what do some of you have to say to the following on analycity:

"In calculus it is seen that functions such as cosx and ln(x-1) can be represented by power series by expansions in either Maclaurin or Taylor series. It is said that these functions are analytic."

So basically this is my original question. Is this a strong definition (a reasonable definition for someone who has only taken up to diff eq)? Without getting wrapped up in complex numbers, whether or not it is synonymous with holomorphic, etc.

Yes, it's good to get back to your original question! Sorry we got off topic. And my original response was that that's a perfectly good definition, provided you make it clear that "represented by expansions in either Maclaurin or Taylor series" mean "is equal to the sum of the Maclaurin or Taylor series".

Of course, for a complex function of a complex variable that is true if and only if the function has continuous derivatives. Also if and only if it satisfies the Cauchy-Riemann equations. And those are easier to check.

 

[Jex]

Yes, you are the only one that actually makes sense to me, lol. Maybe in a couple of years I can engage in the rest of this discussion but for right now your explanation is completely fine with me. It's directly to the point and above all, clear and easy to follow.

Thanks again.

 

[mathwonk]

holomorphic refers only to functions defined on an open subset of the complex numbers and it means they have one derivative.

analytic refers to functions defined on either an open subset of the reals or the compelxes, and it means they are define locally by taylor series. this implies they have infinitely many derivatives.

in the real case having one derivative and being represented by a taylor series are different, but in the complex case they are equivalent.