Talk:Calculus/Complex analysis

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Complex analysis looks interesting however I would like to know more. What about complex integration?

[edit] Structure

• I've gone ahead and started a restructuring of the book, with the goal of eventually expanding it to the level of coverage common to an introductory level Complex Analysis class (ie, at least through integration). The outline I'm planning on is more or less the following

Ch.1, Complex Numbers

1.1 Definitions
 1.1.1 Geometric
 1.1.2 Matrix
 1.1.3 Traditional (i=sqrt(-1))
 1.1.4 Relation between them
1.2 Complex Number Field
1.3 Polar and Exponential Forms
1.4 Roots

Ch.2, Complex Functions

2.1 Polynomials
2.2 Exponential
2.3 Logarithm
2.4 Trig Functions
2.5 Riemann Surfaces
2.6 Branches

Ch.3, Topology of the Complex Plane

3.1 Regions (Open/Closed Sets, Un/Bound, etc)
3.2 Limits
3.3 Continuity
3.4 Sequences
3.5 Series

Ch.4, Analytic Functions and Power Series

4.1 Analytic Functions
4.2 Cauchy-Riemann Equations
4.3 Harmonic Functions (maybe... nice connect to analytic though)
4.4 Taylor's Theorem
4.5 Uniqueness of Power Series and Radius of Convergence
4.6 Mermomorphic and Entire Functions
4.7 Analytic Continuation

Ch.5 Integration

5.1 Contour Integration and Cauchy's Integral Formula
5.2 Classification of Isolated Singularities
5.3 Winding Number and Residue
5.4 Laurent Series
5.5 Improper Integrals and Cauchy's Residue Theorem

Ch.6 Advanced Topics (maybe)

6.1 Infinite Products and Functions of Finite Order
6.2 The Possion Summation
6.3 Arithematic Functions and Dirichlet Series
6.5 Conformal Mappings
6.6 Fixed Points of a Fractional Linear Transformation
6.7 The Riemann Mapping Theorem
6.8 Weierstrass's Theorem
6.9 Mellin Transormation
6.10 Elliptic Functions and Liouville Theorem
6.11 The Gamma, Sigma and Zeta Functions
6.12 The Riemann-von Mangoldt Formula
6.13 The Lindelof Hypothesis
6.14 Stirling's Formula
6.15 The Prime Number Theorem

So, who has any comments, suggestions, or complaints? --Notanut 13:39, 8 Mar 2005 (UTC)

• Actually, looking that outline over again, it seems to me that there need to be a section 2.0 on Complex Functions in general before going into special functions - it may even be desirable to break the chapter in two, one on general Complex Functions (things like mappings from C to C, branchs/Riemann Surfaces, and general properties (with a few specific examples)) and a Chapter on Special Functions (basically 2.1 to 2.4 now).

--Notanut 02:29, 10 Mar 2005 (UTC)

Do not duplicate content. We already have a text on complex numbers! Now content in Calculus:Complex_analysis/Complex_Numbers not in Algebra/Complex numbers will have to be merged. (I've done so. 08:15, 10 Mar 2005 (UTC))

Now, with your outline, that may be usable, but it's best to add content in a structured manner rather than add the structure and then work around it. Dysprosia 07:46, 10 Mar 2005 (UTC)

Sorry about the duplication, I'm used to hardcover Complex Analysis books where an opening chapter on complex numbers is basically manditory. That said, the proofs of Euler's formuala and some of the more abstract ideas like the complex numbers as a field are really too advanced to put in a high school level introduction to complex numbers.

Ok, never mind the high school comment - I confused the HSE Complex Numbers article with the plain Complex Numbers article.

I'm sure you're right about the outline, but I just have a much easier time when I'm doing technical writing when I actually have an overall plan. Obviously there a pretty much a null chance the book would actually come out looking like the outline, I just like having one. --Notanut 20:32, 10 Mar 2005 (UTC)

Perhaps I (I'm a physicist) learned complex analyses in a different way than most mathematicians, but I suggest to move the integration stuff forward. After you derive Cauchy's integral formula for complex differentiable functions you can easily show that if you can differentiate a function once, you can differentiate it twice, so the function is infinitely often differentiable. Also you can show that the Taylor series will converge to the function.

Then you need to introduce the principle of analytical continuation. This can be used to introduce complex analytical functions such as Exp[z] etc.

Then you can proceed with all the usual stuff such as the Maximum Modulus theorem, Residue theorem for meromorphic functions etc. It would be nice to revisit expansion into partial fraction everyone knows from high school analyses using complex analyses.

Then some examples of calculation of integrals using the residue theorem. Also integration of functions with branch point singularity (e.g. by using analytical continuation of function accross the branch cut).

Count Iblis 8 July 2005 23:11 (UTC)

I wholeheartedly agree. Holomorphicity and the Cauchy Integral Formula should be introduced as early in the text as possible. Without them, there's not much to say about power series, analytic functions, meromorphic functions, singularities, or analytic continuation beyond the definitions. --Danvk 18:58, 18 February 2006 (UTC)

[edit] Added some integration material

I've gone ahead and added some material and examples relating to complex integration and Cauchy's theorem/integral formula. No proofs yet, but the examples are fairly complete.

Also, I'm new to wikibooks, and I'm a little turned off by the fact that only about one in five of my edits actually gets saved. The rest of the time it says I've "lost session data" and need to try again later. Any idea what's going on? I'm using Firefox on Windows, so this probably isn't a browser issue... --Danvk 23:08, 9 February 2006 (UTC)

Hi Danvk, nice edits! About equality of functions (analytic continuation), I think it is enough to demand that f and g are equal on a sequence of points with a limit point in the region. The proof consists of constructing the power series about that limit point, which then turns out to be the same for both functions. Let's see if this comment gets saved :) Count Iblis 23:24, 17 February 2006 (UTC)

After having the same thing happen to me on en.wikipedia.org, I figured out that it was actually a firefox extension issue. I removed the extension, and have had no problems since. You're correct -- there only needs to be a sequence of points that converges, but I've never seen a non-contrived example where the disc condition wasn't sufficient. I think saying that equal on a disc implies equal everywhere in the region of mutual holomorphicity makes it much more intuitive why this is a remarkable fact. --Danvk 18:49, 18 February 2006 (UTC)

The sequence of points is useful to show that analytical continuations of real functions are unique. If you have some real function f(x) and if you have two analytical functions g(z) and h(z) that are continuations of f(x) then g and h must be the same function.

Also, in theoretical physics you sometimes encounter analytical continuations of functions defined on the integers, e.g. d-dimensional integrals being continued to arbitrary d. But this isn't unique unless you make some assumptions about the asymptotic bahavior.Count Iblis 23:00, 18 February 2006 (UTC)