# Talk:Complex logarithm

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## History

I don't have a reference in front of me, but at some point it would be nice to include a History section in this article describing the original debate over logarithms of negative numbers. Before Euler figured it out, some intelligent men made some amusing mistakes, like forgetting about integration constants. Melchoir 21:33, 30 July 2006 (UTC)

Women too... —Preceding unsigned comment added by 132.206.33.67 (talk) 21:55, 3 October 2007 (UTC)

## Cut and paste

I dug back through the history on this page, and discovered that somebody named TAB did a cut and paste job from the natural logarithm page to get this thing started. Here's the permanent link if you're interested.

Anyway, I guess that accounts for the rather abrupt introduction as the article stands right now. I'm aiming to regularize the language a little bit, and maybe split it up into sections, so it will look more like a regular Wikipedia article and less like a cut-and-paste job from another article. DavidCBryant 20:39, 16 January 2007 (UTC)

I've thought about it a little more, and have formed a game plan.

• Explain the principal branch as the analytic continuation of ln(z) into the complex plane.
• Explain the principal branch in terms of the exponential function.
• Explain the multi-valued logarithm, and tie it in with the corkscrew Riemann surface.
• Clean up the expressions for log(z) in terms of x + iy.
• Add a history section, if I can get that figured out well enough. I like Melchoir's suggestion.

I'm probably talking to myself here, but this outline may eventually prove useful to somebody else. DavidCBryant 21:14, 16 January 2007 (UTC)

The first cut of this article is now completed. Thanks are due CMummert for his eagle eye, and for the lovely image of the "corkscrew" Riemann surface. After some reflection, I decided to omit the conversion formulae from Cartesian co-ordinates, since these are covered adequately in many articles.

I now intend to add one more section, the complex logarithm as a conformal mapping. I think this will round out the article nicely (except for the history section, which would also be nice – facts are needed to do a good job on that). DavidCBryant 13:46, 22 January 2007 (UTC)

## Is log (0) defined?

It states that log (0) is not defined, but by just looking at the two graphs illustrating the real and imaginary parts of log and extrapolating the real part, would it not be fair to say that log (0) is equal to negative infinity real part and zero imaginary part? Its just a thought, maybe that means that it isnt defined?

Your use of the word "infinity" does, indeed, show that log 0 is not defined. — Preceding unsigned comment added by 86.132.223.160 (talk) 12:54, 19 August 2016 (UTC)

Also if you think of this: xy = 0; where y = - infinity; 1/(x^infinity) = 0; therefore log (0) inverably = - infinity; —Preceding unsigned comment added by 81.129.146.7 (talk) 22:34, 16 February 2008 (UTC)

Can "negative infinity real part and zero imaginary part" really be taken as the definition of the value of a function? I don't think so. But even if it can, it's not the answer. Consider approaching zero from below; the answer would then be negative infinity real part and pi imaginary part. One thing you can be sure of is that 1/log(x) has a limit at 0 for x approaching zero. Dicklyon (talk) 05:16, 17 February 2008 (UTC)
On that note, is the principal branch of the log function defined for -1 (or any other with arg = pi) ? —Preceding unsigned comment added by 200.181.89.3 (talk) 00:59, 24 May 2008 (UTC)
Yes, you just gave the value which is . arg for the principal value is in the open-closed interval from minus to plus pi. Dmcq (talk) 16:36, 19 August 2016 (UTC)

## Is imaginary base logarithm at all notable?

I think I'd like to see some reference to a use of imaginary base logarithms outside of this wiki. I found a reference to the more general idea of a complex number base logarithm on JSTOR so that might be worth a paragraph but all the formula here are I believe unwarranted. Dmcq (talk) 12:44, 25 October 2008 (UTC)

## Proposed revision

A proposed revision of the complex logarithm page can be found at my user page User:FactSpewer, with explanations of the proposed modifications. The main goal is to make it more mathematically precise and informative. Also, Dmcq's criticism of the imaginary-base logarithms section is a good one, so the new draft omits this section, though this can be restored if there is ample evidence that it should be.

Comments are welcome! Let's put them in this section of Talk:Complex logarithm. Thank you, FactSpewer (talk) 17:28, 25 October 2008 (UTC).

I've had a quick look and overall my reaction is negative. There are some good bits which could be moved in but it also removes good bits. I think it is too big a change as it is. It needs to be structured into additions, deletions and rearrangements all done separately and separately discussed. The deletions in particular need to be done carefully. As it is it is very difficult to see what exactly is being removed. I don't think I can comment on it properly as it is. Dmcq (talk) 18:17, 25 October 2008 (UTC)
I understand your request, but it is a little hard to do what you are asking for, since fixing the definitions at the very beginning necessitates changes throughout the article; making changes piecemeal would lead to a lot of inconsistencies (not that the current version is consistent, when it comes to specifying whether log(z) denotes the principal value or an arbitrary logarithm or the set of all logarithms or ??? !) Anyway, I will try to summarize the main changes. I hope to hear what in particular you and other readers like or don't like.

• Precise definition of logarithm
• Precise definition of principal value of argument and of logarithm
• Precise definition of branch of log z
• Precise definition of branch cut
• A little more detail about differing notational conventions.
• Construction of branches using contour integration
• The precise relationship between the Riemann surface and branches

### Deletions

• The entire section on "Derivation from the Taylor series"; I don't know any text that introduces log this way, but feel free to provide one if you know one.
• The imaginary-base logarithms. This could be kept, but I would not recommend it.
• Some of the material in the conformal mapping section. Again, some of this could be kept, though it is a little wordy for my taste.

### Changes

• Using the notation Log z to distinguish the principal value from other logarithms, following Sarason's text.
• In general, rewriting to clarify what is being stated for the principal value, and what is being stated for arbitrary branches of log z.
• The section on "log z" as a multivalued function is clarified by introducing the multitude of logarithms of a number right at the beginning of the article, and by making precise the relationship between the Riemann surface and branches.

See my user page for other details and rationale.

--FactSpewer (talk) 21:59, 25 October 2008 (UTC)

Okay a few comments since you seem unwilling to split up your edits. Spliting them up would enable the diff to point out problems easier for others. Wiki is a collaboration, its best to do things in small chunks except for entire new sections or where things are obviously very badly wrong - which I don't believe is the case for this article.
This is not a textbook, it is an encyclopaedia. One can be a bit less than precise if it makes it easier to read. They should look up the references if they want to know more. If they are given things absolutely precise from the start they won't want to look up more.
The introduction bit should be more chatty like the original.
People like series. If they are notable and relevant they should be kept in. They make an article pretty.
If a big section of work that someone put good faith work into is being chucked out you should explain your reasons in a separate section on this talk page and have a go at contacting the author on their talk page.
Dmcq (talk) 09:20, 27 October 2008 (UTC)
I agree with Dmcq to some extent. Particularly, the old lead was more in line with wikipedia style. It should be no problem to include a precise definition in it while keeping the more introductory style. The rest doesn't look bad, but moving somewhat incrementally is still a good idea. The Taylor series can be left in, but probably not as a way to derive the analystic continuation unless someone finds a source that does it that way. Dicklyon (talk) 15:46, 27 October 2008 (UTC)
In analytic continuation the complex logarithm is given as the example of how to derive the analytic function from a series valid in a small region. Dmcq (talk) 16:11, 27 October 2008 (UTC)
OK, but we should find an actual source. Wikipedia articles are never acceptable as sources. Dicklyon (talk) 16:27, 27 October 2008 (UTC)
Found an example in Ablowitz, Mark J.; Athanassios S. Fokas (2003). Complex Variables Introduction and Applications. Cambridge University Press. p. 154. ISBN 0-521-53429-1. Cite uses deprecated parameter |coauthors= (help). But having looked at the article that section becomes a bit lost after the first two series, it isn't really an example of analytic continuation. Dmcq (talk) 17:21, 27 October 2008 (UTC)
It doesn't seem to show the Taylor series. Dicklyon (talk) 18:31, 27 October 2008 (UTC)
Impressive, I must use the web more to find things. I thought it using the complex logarithm as an example of analytic continuation and mentioning the Taylor series was enough. I'll try and find an example with the series shown explicitly. Recently I tried to find something in a book when I knew it was in there but I just couldn't find it at all - very frustrating. Dmcq (talk) 20:03, 27 October 2008 (UTC)
How about Gronwall, T. H. (1916), "On the Power Series for log (1 + z),", The Annals of Mathematics., 18 (2): 70–73 at [1] Dmcq (talk) 20:51, 27 October 2008 (UTC)

That's not a bad reference for an example of analytic continuation; it's written much more coherently than the section in the current Wikipedia article. On the other hand, I don't think that this author (or the author of any complex analysis textbook I know) is claiming that analytic continuation is the appropriate way to introduce the complex logarithm. If you continue reading beyond the first page of that article (I looked it up), you will find that it is a somewhat complicated approach. The author's purpose there is to illustrate analytic continuation, not to define log z in the most understandable way.

Not all readers will want this level of sophistication. I think the article should start out more simply. In particular, opening with the principal value may be the easiest for many readers to understand. It gives a quick and correct definition. Following Wikipedia style guidelines, the article can get more sophisticated later on as the article progresses.

I think Dmcq's idea to keep the power series may be a good idea, as long as it's not suggested as a reasonable way to introduce the complex logarithm. Maybe it could be mentioned later on as a definition of a branch of log z defined in the radius 1 circle centered at 1 (that is what it is). I'll try adding this to the draft.

To anyone who thinks there is nothing seriously wrong with the article I would ask: what is the meaning of "log z", according to the current article, for a particular complex number z, like 1+i? Does this notation mean the principal value? Does it mean a set of numbers? Or something else? I find the article unclear on this point. An article on the complex logarithm should be clear on this point!

I agree that it is possible to go overboard with precision. We should strive for the simplest explanations that are meaningful and correct. I don't believe in sacrificing correctness, however. Readers of a Wikipedia article on mathematics are (I would guess) looking more for correct information than for entertainment, more often than not.

Making the intro more chatty is a good idea. I'll try doing this. It should be possible to do this without sacrificing correctness.

As for making edits piecemeal, maybe you are right that that is the way to go, even if it introduces inconsistencies along the way.

Thank you all for the input. --FactSpewer (talk) 02:39, 29 October 2008 (UTC)

I just posted a new version on my user page. It restores the Mercator power series for log(1+z), and mentions the Gronwall paper discovered by Dmcq. And it makes the intro more chatty (while being careful to remain honest).
By the way, I was thinking that in the Principal Value section, we could possibly remove the first sentence and the next paragraph. Do you think this would make it easier or harder to read? I'm not sure, so your advice would be helpful. --FactSpewer (talk) 03:32, 29 October 2008 (UTC)
You have a real problem with chatty. The original intro is far better. How about just giving up on trying to rewrite that? I get the feeling you are far more interested in a one true way to complex logarithms which must be wrapped up tight like a legal document. I see the encyclopaedia as talking to people who have any different needs and most will look at this to just introduce the subject and give them enough to be going on with. If there was a way of giving them a spiral to run their fingers on I'd be happy because some people 'get' things better that way. 10:36, 29 October 2008 (UTC)
Dear Dmcq (please sign your Talk posts): I am trying in good faith to improve the article, to make it clear and correct, and to have it reflect the published literature. If you think there are other more understandable approaches to complex logarithms than the approach beginning with the principal value and other branches, please suggest them (and if you can, find published introductions to complex analysis that introduce them this way to support your position). If you think it is not important that the article attach a clear meaning to the expression "log z" when it appears in equations, then please say so. By the way, my draft covers only a small fraction of Sarason's treatment.
May I suggest that you prepare your own draft showing what you think the article should look like, and explaining what references you are following in developing it? This would make it a lot easier to discuss the specifics of the content. --FactSpewer (talk) 06:37, 30 October 2008 (UTC)
I think the current version of the introduction is straightforward and to the point. It doesn't need much justification for notability. It explains what the major problem is with branches and doesn't go round assuming only the single branch version is being defined. It defines the log succinctly and even says how to spot it when written, and it does this all within a few readable sentences. I'm happy with it as it is. Dmcq (talk) 10:40, 30 October 2008 (UTC)
The current version does not mention that the complex logarithm is the "inverse" of the complex exponential. I think that's the best way to introduce it. It mirrors how you would introduce logarithms in the real setting. In that sense, the draft is better. Unfortunately, there are some points I don't like about the draft. Most importantly, the style (too many lists, too formal) hinders intuition; that's the same point as Dmcq. The draft talks too little about the multi-valued approach and perhaps too much about branches (other functions like the square root also have branches; perhaps Sarason talks a lot about branches because it's the first time they appear in the book?). I do agree that the section on imaginary bases should be cut down dramatically. -- Jitse Niesen (talk) 12:36, 30 October 2008 (UTC)
The current introduction doesn't say it is the reverse of exponentiation but it is implicit in that it gives the polar form z=re and then gives log(z) in terms of that. The exponential function article will need a look at as it starts defining itself in terms of the logarithm! Historically I suppose it started as the anti-logarithm. Dmcq (talk) 12:50, 30 October 2008 (UTC)
Dear Jitse, thank you for the feedback. I agree with you that what makes the complex logarithm important is primarily that it is an "inverse" of exp. The fact that some of its branches extend ln is secondary. The formula for log in terms of polar form is also secondary, being motivated by the attempt to find an inverse for exp, but maybe it is worth keeping in the intro in some form instead of moving it further down. As for the style, I will think about how to rewrite the body of the article (i.e., the part after the table of contents) in a more friendly way without losing the content. --FactSpewer (talk) 03:38, 31 October 2008 (UTC)

## Logarithms to other bases

Hi, everybody. Since one thing that we seem to agree on is that the section on imaginary-base logarithms is far too long given its relevance, I am going to revise it by generalizing it to arbitrary bases and removing the focus on the case where the base is imaginary. I'm not sure if the example is worth including. Actually, I'm inclined to remove the section altogether, but maybe keeping this small version of it is a reasonable compromise. Happy Halloween, FactSpewer (talk) 02:54, 31 October 2008 (UTC)

## Introductory section

The introductory section has been changed I believe from something which was mostly right to something which is just wrong. It used to say:

In complex analysis, the complex logarithm is the extension of the natural logarithm function ln(x) – originally defined for real numbers only – into the complex plane.

Now it says

In complex analysis, a complex logarithm function is an inverse of (a piece of) the complex exponential function, just as the natural logarithm ln x for positive real numbers x is the inverse of the real exponential function ex.

I had a little problem with he original in that the natural home of the complex logarithm is a Riemann surface not the complex plane. It also glosses over the business about principal value.

The second one however is even further wrong I feel. It restricts the complex logarithm to he principal value which is at best a very old fashioned idea. It is also clunky with is (a pice of) and indefinite articles and the unnecessary bit about positive reals.

I objected to this change but it has gone in anyway. Just reverting would be edit war. How do other people feel about this change? Dmcq (talk) 09:26, 2 November 2008 (UTC)

Any such substantial change in definition needs to be accompanied by a source, at minimum. Then we could start to discuss which definition is more conventional. So I reverted it. Dicklyon (talk) 13:53, 2 November 2008 (UTC)
I think we should not try to put a mathematical definition of the complex logarithm in the first sentence. The question should be: what is the best way to introduce the complex logarithm. And, as I argued above, I think the best way to introduce it is as the inverse of the exponential. If you want a reference, that is how Ablowitz and Fokas, Complex Variables, p. 48 do it. I note that the former definition did not have a reference, so it does seem a strange reason to revert.
I also disagree that restricting the complex logarithm to the principal value is very old fashioned. While indeed the most natural home of the complex logarithm is a Riemann surface, many people will not see it that way. I did not learn Riemann surfaces in my mathematical studies.
However, Dmcq is right that the new introduction is a bit clunky. So let me try to combine both.
"In complex analysis, the complex logarithm is an inverse of the complex exponential function, in the sense that log(z) = w if ew = z. It extends the natural logarithm function ln(x) – originally defined for real numbers only – into the complex plane.
What do you think of that? -- Jitse Niesen (talk) 16:36, 2 November 2008 (UTC)
That sounds much better. It doesn't make a big decision over whether it is really the multivalued version or a single branch or a Riemann surface or whatever, and I like that too. And more to the point it just sounds to me more like an introduction, rather than some treatise halfway along the route to being polished into complete impenetrability. Dmcq (talk) 18:19, 2 November 2008 (UTC)

Dmcq, I wasn't aware that you thought the sentence was wrong. What you wrote earlier was that it was already implicit in the formula in terms of polar form, and hence didn't need to be included. So I thought you accepted it, but thought it was too obvious to state. If I had known you had a serious objection to the sentence, I would have tried to convince you of its correctness, as I will now try to do.

There are at least three ways to resolve the problem of the complex exponential function not having an inverse in the usual sense:

• View the inverse as a multivalued function, taking each nonzero complex number to the set that is its inverse image under the exponential function. This function takes complex numbers as inputs and outputs not a number, but a set.
• View the inverse as a function from a Riemann surface to the set of complex numbers. This gives a well-defined function, but it does not take numbers as input; it takes points on the surface as input. Whether this is the most natural approach depends on what you want to do with it.
• Restrict its domain to a region on which the exponential function is 1-to-1, so that it has an inverse. This is the approach that leads to the principal branch and other branches. This is the only way that gives bona fide functions from complex numbers to complex numbers.

I think that the article should mention all three of these, and my sentence is intentionally compatible with all three:

• The multivalued function is the inverse of the exponential function as a whole, in the multivalued sense.
• The function on the Riemann surface is the inverse of the exponential function lifted to a map from C to the Riemann surface.
• Each branch of log z is the bona fide inverse of a restriction of the exponential function. (Dmcq: I am not sure why you think that my sentence is singling out the principal branch. If you still believe this, let me know, and I can explain in more detail why this is not the case.)

Dmcq: I am not sure why you keep insinuating that branches are outdated, and that viewing log as a function on the Riemann surface is modern. The Riemann surface approach is in Riemann's papers of the 1850s. Branches of log have been around a long time too, but they are still present in every introductory text to complex analysis I know: they are the standard approach to defining single-valued holomorphic functions of a complex variable with the desired properties. Have you heard of Serge Lang? On p.120 of the fourth edition of his text Complex Analysis (1999), his definition of log is in terms of branches. Have you heard of Donald Sarason? His textbook is from 2007. I challenge you to find a single introductory complex analysis textbook that introduces log for the first time via Riemann surfaces. (I don't mean to denigrate Riemann surfaces; they have their uses. But to suggest that they should replace the approach with branches is ridiculous.)

Dmcq: I am not sure why you wrote that changing "real numbers" to "positive real numbers" is unnecessary. It is not standard to define the real natural logarithm function ln x at 0 or at negative numbers.

The words "(a piece of)" are needed for the definition to be compatible with the various usages of log z. Without this, the sentence suggests the incorrect statement that an actual inverse of e^z exists. Unfortunately, writing "log(z) = w if ew = z" is misleading too, since it suggests the incorrect statement that a function log taking (nonzero) complex numbers to complex numbers with this property exists.

My sentence is a little bit vague in that it does not specify what is meant by "a piece". I agree with Jitse here that it's not necessary to include a mathematical definition in the first sentence, but I do think we owe it to the reader to try not to suggest anything incorrect or misleading, even if it means including a few more words.--FactSpewer (talk) 07:28, 3 November 2008 (UTC)

A few more comments about differences between my version and Jitse's version of the sentence: It is better to say "a complex logarithm" than "the complex logarithm" since there are many different branches of log z, taking different values at any given z. As you probably know, saying that a complex logarithm extends the natural logarithm on the (positive) real numbers implicitly excludes many branches of log z, some of which are not even defined at positive real numbers, and some of which are defined on the positive real numbers but do not agree with ln x there!--FactSpewer (talk) 07:40, 3 November 2008 (UTC)

I said I didn't like your version of the introduction before you put it in. I believe I was unusually forceful in how I said it and gave some reasons so there's no reason to act surprised when I object as it is put in. The introduction is not supposed to be rigorous. Please read Wikipedia:Manual of Style (mathematics). It isn't all that long. I believe it gives good guidelines and it has a nice little bit about what an introduction should be like. As to your individual points:
Your introduction had 'a piece of' qualifier which says to me exactly what it says, not the whole and therefore a branch. I did not see it as in any way inclusive of the various approaches but I'm glad that you're actually considering them rather than the original idea of following a single textbook. They don't have to all be given equal weight or mentioned everywhere at the same time like some equal opportunity board but they are fairly notable and citeable. I though Jitse's version handled this aspect very well.
The business about positive real numbers was unnecessary in an informal introduction and sounded bad with real number being repeated, and it is unnecessary anyway when talking about the real logarithm as its domain is the positive reals.
I applaud your attempt at informality saying 'a piece' rather than wanting to make it more precise like you said you'd like.
Jitse's version said 'is an inverse' which put in the requisite business about having different logarithms as far as I was concerned. Too many indefinite articles makes it sound too vague and wishy washy. There is a nice bit in an example introduction in that manual of style '... is, loosely speaking, a function from ...'. Which I think illustrates a lot about the mindset needed when writing an introduction.
Hope this helps Dmcq (talk) 22:41, 3 November 2008 (UTC)

I got rid of "a piece of" and instead put "inverse" in quotation marks. I also got rid of the stuff about positive real numbers. And I added the citation.

Incidentally I looked up another complex analysis textbook, Functions of One Complex Variable by John B. Conway. Just like the other textbooks, when he gives the definition of log, it is as a branch. Moreover, on p.39 he writes "log z cannot be defined as the inverse of e^z" so some qualification is necessary -- hence my quotation marks. Is there even one textbook that introduces it by defining it on the Riemann surface?--FactSpewer (talk) 02:59, 4 November 2008 (UTC)

Yes I like that intro. The bit in Conway about cannot be defines as the inverse refers to the complex plane and you'll notice he first of all writes the unqualified logarithm as a multivalued set function and always refers to a single valued logarithm function as 'a branch of the logarithm' or 'the principal branch of the logarithm' rather than as the unqualified logarithm. When he does later deal with Riemann surfaces he refers to the version on the Riemann surface as the logarithm without any qualification. This really goes back to the business about wiki being an encyclopaedia. Conway was writing a textbook and building one thing upon another and getting things right. I don't suppose New Math (song) as satirised by Tom Lehrer is the right motivation but "the important thing is to understand what you're doing, rather than to get the right answer" might be a good thing to aim your writing at as you seem to have quite enough drive for being 'right'. Dmcq (talk) 10:09, 4 November 2008 (UTC)

## Motivational section

I intend to add a new section entitled "An inverse of the exponential function?" at the beginning of the body of the article, to motivate the various approaches to log, before getting into the descriptions of branches and the Riemann surface and so on. (This is in keeping with the mathematical style guidelines suggested by Dmcq, which suggest including some discussion to help explain things more informally at first and only afterward getting into the precise mathematics.) This new section admittedly is redundant with some of the material that follows, but rather than fix everything at once, the plan will be to edit things slowly. The later sections can be fixed up later. --FactSpewer (talk) 04:14, 6 November 2008 (UTC)

It looks very good, thank you. Is there any particular reason why you do not mention multi-valued functions in the motivational section? What about adding something in the section about the principal value that it is defined to be the logarithm with imaginary value in (-pi,pi]? But I like the new version much better than what you wrote before, and what was in the article before. I'm not worried about the redundancy; that will resolve itself at some time. Thanks again for your efforts and for listening to what we have to say. -- Jitse Niesen (talk) 11:29, 7 November 2008 (UTC)
Regarding log as a multivalued function: it actually is mentioned, but in a footnote. I think there is not much more to say about it, beyond what is already said about every nonzero complex number having infinitely many logarithms and giving the formula in terms of polar form. It does not solve the problem of exp not having an inverse, but rather ignores the problem. Many textbooks I have seen avoid talking about log z without specifying a branch, to avoid the confusion that can ensue.
Regarding your definition of principal value: that is an excellent idea. I will try to incorporate it into the section.
--FactSpewer (talk) 01:47, 8 November 2008 (UTC)
May I add a me too to what Jitse Niesen said. I'll have a think about the title with a question mark but yes it's looking good. Dmcq (talk) 15:33, 8 November 2008 (UTC)
Thank you, Dmcq. I'll continue then, and if anyone has suggestions for improvement, please write them here. --FactSpewer (talk) 04:18, 9 November 2008 (UTC)
I've finished adding/rewriting material. Next I plan to remove the redundant parts of the article, namely the sections strictly between the Generalizations section and the Plots section. I think all of the material worth saving from those sections has been incorporated into the earlier sections, with improved explanation. If anyone wants to retain other material, now would be a good time to discuss it here. --FactSpewer (talk) 06:21, 15 November 2008 (UTC)

## Riemann surfaces, multivalued functions, and branches

Most of the material introduced in the recent edit is either misleading or included elsewhere in the article already. I'll go through the edits one by one. Adding the link to the branch cut article is OK. The function given on the Riemann surface is not multivalued, and is not a function that takes complex numbers as inputs, as suggested in the sentence "In this solution, the logarithm has the definition..." The recipe for constructing all logarithms of a nonzero complex number is already given at the very top of the article. Calling a function multivalued does not resolve the indeterminacy. In fact one needs to be very careful with discussions of multivalued functions to make the notation mathematically precise, and as was discussed earlier on the talk page, it is preferable simply to say that a number has infinitely many logarithms and to leave it at that, without defining log z to be a set of values. The continuity of branches is already discussed carefully in the complex logarithm#Branch cuts section. If there is no further discussion, I will revert the edit, keeping only the added link to branch cut. FactSpewer (talk) 12:42, 20 January 2009 (UTC)

I implemented the partial revert mentioned above. FactSpewer (talk) 16:09, 25 January 2009 (UTC)

I didn't follow you completely, do I need to motivate my edits here or else they will be reverted? I didn't realize you couldn't write as I did about the Reimann surface thing. --Kri (talk) 22:28, 25 January 2009 (UTC)
Dear Kri, I'm sorry if my message was unclear. I'd be happy to try to explain any particular point; maybe the best place to continue the discussion would be on my Talk page, unless there's something that you think needs to be discussed more broadly. It's just that there has already been a lot of discussion leading to the current version of the article, and in particular on how to treat the multivalued nature of log carefully. Your edits here won't necessarily be reverted, but because the complex logarithm article has been worked on quite extensively already, it might be a good idea to discuss any significant changes on this Talk page in advance if you want to be sure that there is consensus about them. There are probably other articles that are more in need of your help. Best, FactSpewer (talk) 23:30, 25 January 2009 (UTC)
I see, and thanks for your reply. No, your message were not unclear, I hasted through it so it was probably that. For me it doesn't matter if the edits I made are reverted or not; I simply thought it didn't look quite complete and filled the rest of the info in. If someone thinks the older version looks better, that one can gladly change it back. --Kri (talk) 03:23, 27 January 2009 (UTC)
OK, thank you. FactSpewer (talk) 19:05, 27 January 2009 (UTC)

## An inverse of the exponential function?

I assume that the reason for Anonymous Dissident's changing the title "An inverse of the exponential function?" was to avoid the question mark, and also to clarify that the exponential function referred to the complex exponential function. These are reasonable objections, though I am going to reword the suggestion "Problems with an inverse complex exponential function" slightly. This is a nitpick, but "an inverse complex exponential function" is not the same as "an inverse of the complex exponential function"; moreover, "Problems with an inverse complex exponential function" presumes that an inverse exists, whereas the problem lies actually in the fact that it does not exist. --FactSpewer (talk) 07:42, 18 August 2009 (UTC)

No problem. I was, indeed, merely trying to avoid the question mark. —Anonymous DissidentTalk 08:05, 18 August 2009 (UTC)

## incorrect graph

One of the graphs shown at the bottom of the page, labelled z = |Im(Log(x + iy))| cannot possibly be right! It fails to show the discontinuity of 2\pi i across the branch cut!. The correct graphs are shown higher up, in screw-shape form. 67.198.37.16 (talk) 05:23, 1 December 2014 (UTC)

As x+iy approaches the negative real axis from above, Im(Log(x+iy)) tends to π. As x+iy approaches the negative real axis from below, Im(Log(x+iy)) tends to -π. So the absolute value of Im(Log(x+iy)) is continuous on the whole plane except at the origin where it is undefined. So I think there is no problem with the graph. Ebony Jackson (talk) 02:06, 16 December 2014 (UTC)
Though the graph fits the caption, taking the absolute seems rather pointless (and unnecessarily confusing). The first time I saw it I did a double-take. —Quondum 03:42, 16 December 2014 (UTC)
I would say delete that whole section and all its graphs. Not useful at all. Dicklyon (talk) 06:59, 16 December 2014 (UTC)
Agreed. The only graph of interest might be one like the leftmost, to supplement the plot of the imaginary part of the multivalued log function in the first section. —Quondum 14:24, 16 December 2014 (UTC)
Actually I agree too. The only one possibly worth keeping is the leftmost one. But if it were up to me, I'd probably delete all of them. Ebony Jackson (talk) 09:38, 25 December 2014 (UTC)