Talk:Complex analysis

Applied math

It seems to me that many schools each complex analysis in the applied math department. I notice that the article does mention applied math, but only down the list. Is it possible to include the reason why it is in AMa? Gah4 (talk) 06:42, 3 March 2021 (UTC)

Gah4 Does it work if we explain it based on Hermann Weyl's achievements? Perhaps it is also useful for several complex variables. I'm not ready to write it.--SilverMatsu (talk) 02:11, 24 June 2021 (UTC)

It seems that there are classes that are more for theoretical math, and those more for engineers. I was thing about something like ACM95 at Caltech, which is required for most (or all) engineering degrees. I believe other schools have a similar course, often part of the AMa department. Gah4 (talk) 07:35, 24 June 2021 (UTC)

Gah4 Thank you for reply. I think complex analysis is related to AMa because it is related to vector analysis and harmonic functions, but I'm not ready to write it. And you can edit this article directly. Previously it was B-Class, but now it is C-Class. Therefore, there is nothing wrong with reorganizing the section.--SilverMatsu (talk) 08:56, 24 June 2021 (UTC)

(edit conflict)"Applied math" has at least four different meanings that are related but distinct. It may be the name of some courses or some university departements. This is not relevant for an encyclopedy, except for being mentioned in Applied mathematics. "Applied math" may refer to mathematics that are used outside mathematics. It appears that almost all parts of mathematics are implied (most items listed in APM95 do not belong to complex analysis), and this is why many of our articles have a section on applications, generally near to the end of the article. "Applied math" may refer to mathematics that are drived by applications. Finally "applied math" is often used as an alternate name for numerical analysis.

Yes. Well, it probably could apply to all mathematics used in sciences such as physics, and most engineering. For traditional reasons, it often doesn't. The AMa departments in schools that I know do things like computational fluid dynamics. Mostly numerical, so pure math people aren't so interested. I suspect, yes, it is sometimes used for numerical analysis but more often used for the less theoretical cases. And yes it is used for names of courses taught by people in the named department. So, yes, connected but distinct. Gah4 (talk) 09:39, 24 June 2021 (UTC)

The applied complex analysis, such as Laplace transforms used for EE filters, and contour integrals for a variety of computations, including inverse Laplace transforms. Also, some definite integrals that are hard to do other ways. Gah4 (talk) 09:39, 24 June 2021 (UTC)

In view of this, it would be useful to add a section "Applications" near to the end of the article, but this requires an editor with the needed competences and/or a reliable source on this. I do not see anything else to do. D.Lazard (talk) 09:21, 24 June 2021 (UTC)

I am not sure I am quite good enough to do it, but I will think about ones that I can add. Gah4 (talk) 09:39, 24 June 2021 (UTC)

I am considering to explain the relevance to Applied math by extending the exposition for Cauchy–Riemann conditions, mentioning the harmonic function, and then the Laplace equation, but I am not ready for reference and details. Next, we need to consider whether a holomorphic function actually exists, but for the open Riemann surface there is an Behnke–Stein theorem.--SilverMatsu (talk) 13:20, 24 June 2021 (UTC)

i happened to stumble upon this interesting exchange and i agree with the above: an applications section would definitely be handy. i also concur with D. Lazard that it would also require some proficiency in the area. it would not be fun, and would probably require tying the material from a few numerical analysis textbooks with the appropriate source for the "pure" variant. * dibbs out. maybe i'll lift someone else's edits and try to make it better. one suggestion i have is to motivate the section using the discussion of ordinary differential equations, particularly in terms of complex roots, move to fourier's (inferior) transform before discussing laplace. we have to start with DEs in my opinion. it will make everything easier. i never saw the use of complex analysis (being a huge fan of real analysis) until i started looking at the theory of differential equations. then i grew an appreciation for complex theory that is only starting to blossom, ten years after getting a reasonably-good grasp on real analysis theory. good chat here lads. "198.53.108.48 (talk) 19:04, 24 June 2021 (UTC)

@Gah4 and D.Lazard: I found reference. Is it appropriate for this case?--SilverMatsu (talk) 15:29, 26 June 2021 (UTC)

Yes that sounds about like I meant. Gah4 (talk) 01:21, 27 June 2021 (UTC)

Reorganize the section

Suggest to reorganize the section. First, I'm thinking of merging the holomorphic functions and complex functions sections. Next, I'm thinking of putting a section about the complex numbers (complex algebra) and the properties of the complex plane. I'm still thinking about the arrange of the sections, so I haven't thought about the details of the contents of the sections yet.　--SilverMatsu (talk) 05:29, 20 October 2021 (UTC)

It would be good to provide a compressed summary of at least the main sections of an introductory complex analysis textbook, at least briefly describing the major ideas and results (the current "major results" section is much too compressed, and should be split into pieces). It might start with a quick explanation of complex numbers (describe how they add and multiply, and the negation, reciprocal, and complex conjugate), then talk about holomorphic functions, harmonic functions, power series, Laurent series, zeros and poles, line integrals and the residue theorem, analytic continuation, conformal mapping and the Riemann mapping theorem, elliptic functions, Riemann surfaces, uniformization, Fourier analysis, maybe the Riemann zeta function, .... It’s okay if some material is duplicated from other articles and it’s also okay if details and proofs are omitted and descriptions are more expository than formal/technical. It would be great to include as many figures as possible, because top-level articles like this are likely to have many nonspecialist readers. Disclaimer: I am not a mathematician. –jacobolus (t) 16:36, 20 October 2021 (UTC)

Thank you for your reply. Now that we've started creating the list, please make update and additions. I agree that we need to add a figures. Thanks to @Nschloe: for creating the figures.--SilverMatsu (talk) 03:15, 21 October 2021 (UTC)

section list (draft)

• History
• Holomorphic functions
  • Cauchy–Riemann conditions
  • Cauchy's integral formula
  • Analytic functions (Power series (Laurent series))
• Meromorphic function
  • Properties of singularity and residue (Especially isolated singularity)
• Harmonic function and subharmonic function
• Analytic continuation and Riemann surface
  • Conformal mapping and the Riemann mapping
• Application
  • Analytic number theory

Difference between "Series" and "Sum of a series"

An user (User:Jacobolus) reverted my edit concerning the difference between a series (of functions) and the sum of such a series. In my understanding, these are not synonyms.

• A series or a sequence refers to a mathematical object related to a sequence of terms (numbers or functions), or a sequence of partial sums : "In modern terminology, any (ordered) infinite sequence {\displaystyle (a_{1},a_{2},a_{3},\ldots )}{\displaystyle (a_{1},a_{2},a_{3},\ldots )} of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other."
• On the other hand, the sum of a series (or, correspondingly, the limit of a sequence) refers to a number (or a function), namely the limit of the sequence of partial sums : "In this case, the limit is called the sum of the series."

See also this page for instance: https://www.uio.no/studier/emner/matnat/math/MAT2400/v11/RealAnalCh3.pdf : "We shall be particularly interested in how general functions can be written as sums of series of simple functions such as power functions and trigonometric functions. " --L'âne onyme (talk) 16:46, 27 October 2021 (UTC)

Besides, I think the article on Taylor series that you quoted was also incorrect, so I corrected it. --L'âne onyme (talk) 17:01, 27 October 2021 (UTC)

You may be thinking about a formal power series. But a complex-differentiable function necessarily has a converging Taylor series. The common usage is for the words "series" or "Taylor series" to mean the sum of terms; saying "sum of the Taylor series" seems unnecessary. Maybe someone who is a professional complex analyst can chime in. –jacobolus (t) 17:37, 27 October 2021 (UTC)

Yes, they do have (normally) converging Taylor series (in closed sets inside the open disk of convergence) but I don't see how that pertains to the subject. The sum of a series is still not synonymous with the series. If it is a common usage to use one word as a synonym for the other (which I am not aware of), it is still better to distinguish between them in Wikipedia for the sake of clarity. L'âne onyme (talk) 17:49, 27 October 2021 (UTC)

This is not “use one word as a synonym for another”, but rather “the same word has different definitions depending on context”, and there is no requirement to pedantically force the use of your preferred definition across Wikipedia. –jacobolus (t) 01:43, 28 October 2021 (UTC)

When people write down the “Taylor series” e.g. ${\textstyle \sum _{n=0}^{\infty }f^{(n)}(0)x^{n}/n!,}$ that name and expression both typically refer to the sum. That is what the ${\textstyle \sum }$ means. At least, in my understanding. Maybe you can do a survey of introductory complex analysis textbooks and figure out what proportion of them insist consistently on this distinction. –jacobolus (t) 01:47, 28 October 2021 (UTC)

Whilst I agree that the definition might be clear from the context, it is certainly not "pedantic" to write that "a function is the sum of its Taylor series" (and by the way, please remember to remain polite). To say that a function is equal to a series is not only mathematically incorrect (a series being a mathematical object related to a sequence), which would be acceptable if it were to avoid pedantry, but it is also a very bizarre formulation. As for your suggestion to do a survey, I've already offered a link above, and here are some more :

• [1] The term "sum" is consistently used, see p.10 "the sum is continuous on each such interval" ; "A function that is the sum of a power series...", p. 12 "It turns out that the sum is also continuous at the endpoint"...
• [2] p.122 "its sum is a continuous function"
• [3] p.2 "the function f is called the sum or the point-wise sum of the series"
• [4] p.81"the function f is the sum of the series"
• [5] p.5 "Then the series [...] is pointwise convergent on (−1, 1) with sum" ; "the sum of the series is discontinuous" ; p.6 "The sum function s(x) of the series" ; p.16 "the sum function"...

Now it's over to you to give me an example of a work that uses the term "series" as a synecdoche for sum. --L'âne onyme (talk) 00:21, 1 November 2021 (UTC)

I just skimmed through 4 complex analysis textbooks by fastidious authors. 2 of them never explicitly talk of the “sum” of a series (only partial sums which converge to some function), 1 of them occasionally uses the formulation “sum of a series”, and 1 consistently uses the language “sum of the series” (Wegert’s Visual Complex Functions, which takes it to the extreme of e.g. Any function f : D → C which is analytic in a ring domain can be uniquely represented as (the sum of) a Laurent series”), it seems they generally make statements along the lines of “f is representable by a power series Σ...”, “f has a power series expansion Σ...”, “f can be expanded in a series Σ...”, or “the series Σ... converges to f”. You may be right that the language “a differentiable function of a complex variable is equal to its Taylor series” is a bit sloppy. –jacobolus (t) 07:07, 1 November 2021 (UTC)

On the other hand, Google scholar finds 143 papers which use the exact phrase “equal to its taylor series” (and also finds 25 papers with the exact phrase “equal to the sum of its taylor series”). For example Boaz (1989) "When Is a C^∞ Function Analytic?" The Mathematical Intelligencer 11(4). “Is such a function real-analytic (equal to its Taylor series), or might it have a singularity of the second kind?”. –jacobolus (t) 07:42, 1 November 2021 (UTC)

Ok, I guess you have convinced me. --L'âne onyme (talk) 09:16, 1 November 2021 (UTC)

I think you also convinced me that we could reword to be more precise. Most careful complex analysis textbooks seem to be fairly consistent in using "series" to mean formal power series (introductory calculus textbooks tend to be a bit less careful/formal). Skimming more sources it seems that this "sum of a series" language is one relatively common approach to distinguishing a formal power series from its interpretation as a function; I had not payed attention before, but I can see how people who have read a few books which use that language consistently would get used to it and come to expect it. Instead of “equal to” some phrasing like “can be represented as” might be more precise. I wonder if some conventions should be picked for use in series (mathematics), power series, Taylor series, Laurent series, etc., and throughout math wikipedia, to avoid ambiguity. –jacobolus (t) 17:36, 1 November 2021 (UTC)

Regarding equal. see Identity theorem.--SilverMatsu (talk) 09:38, 3 November 2021 (UTC)

I do agree that one should distinguish a series (which does not necessarily converge) from the sum of the series (which does not necessarily exist), and also whether one considers a series of numbers or of functions (such as the Taylor series of a function) -- although the notion of series can be defined in abstract topological modules including numbers as well as functions. But in the latter case there is usually a vast choice of different topologies (pointwise, in given points; or uniform, or with respect to some norm on the function space [to be specified, again with many choice possible]) for which the series may or may [and in general will] not converge. A series is (in rigorous math texts defined as) not just a sequence (of terms) but a pair <math>((a_n), (\sum^n a_k)) of two sequences, that of the terms and that of the partial sums, while the sum of the series is the limit of the sequence of partial sums, if it exists -- which almost always [in the sense of measure theory] is not the case. — MFH:Talk 15:41, 6 December 2024 (UTC)