Talk:Complex logarithm

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)

We now have History of logarithms where the following reference is found:
 * Florian Cajori (1913) "History of the exponential and logarithm concepts", American Mathematical Monthly 20: pages 5 to 14, pages 35 to 47, pages 75 to 84, pages 107 to 117, pages 148 to 151, pages 173 to 182, pages 205 to 210, links from Jstor

Though over a century old, Florian Cajori recounts the drama of the development of the multivalued-function. — Rgdboer (talk) 01:07, 11 August 2019 (UTC)
 * Here is Gauss in an 1811 letter to Bessel (Werke, VIII, p. 90) defining log as $\int \frac1x dx$ and discussing it as a multi-function. https://books.google.com/books?id=aecGAAAAYAAJ&pg=PA90 –jacobolus (t) 08:06, 15 October 2021 (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?

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)


 * 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)


 * Yes, you just gave the value which is iπ. arg for the principal value is in the open-closed interval from minus to plus pi. Dmcq (talk) 16:36, 19 August 2016 (UTC)

You can come up with a definition for log(0) if you squint and work up your own non-standard prerequisite definitions, but it doesn’t fit super well with conventions typically used with complex arithmetic. The real part is –∞ on the Extended real number line, while the imaginary part is undefined (or if you like, can take any value). Probably the most straight forward would be to take the complex plane (or a strip with imaginary part in [–π, π), or if you like a cylinder $$\mathbb{R} \times S^1$$) and append two additional points ∞ and –∞ (note this is a different thing than the extended complex plane), and then just declare the that log of 0 is –∞ by definition. –jacobolus (t)

Complex logarithm of a number vs. complex logarithm function
It would be helpful in the top paragraph to disambiguate

1) the notion of "complex logarithm of a nonzero complex number z" (a complex number w such that $e^{&#8239;w} = z$) from

2) the notion of a "complex logarithm function".

It is the latter that is a partial inverse of the complex exponential function. Also, the article is wrong to claim that "the complex logarithm cannot be defined to be a single-valued function" (on $$\mathbb C^\times$$) - it can be, by making choices. What cannot be done is to make the choices so as to make the function continuous everywhere. Ebony Jackson (talk) 23:41, 1 July 2020 (UTC)

The comments above have already been incorporated. Ebony Jackson (talk) 02:15, 30 December 2020 (UTC)

What is missing?
What aspects of the complex logarithm are missing from this article, besides the history? It already seems to cover at least what is in the books by distinguished complex analysts such as Ahlfors and Sarason. It would be nice to have constructive suggestions of sources (by equally distinguished experts) that contain additional relevant material. Ebony Jackson (talk) 06:56, 18 September 2021 (UTC)
 * I have to say I share your question Ebony Jackson. I was a bit surprised by the reclassification of the article, but I have never been an assessor, so I am not familiar with the exact process. BFG (talk) 11:11, 18 September 2021 (UTC)
 * I think of two lacking points:
 * I have added the definition by analytic continuation in the lead. IMO, this deserves to be expanded to its own section.
 * The complex logarithm is a holonomic function. Among other things this implies that the coefficients ot the Taylor expansion at every nonzero point satisfy a linear recurrence relation with polynomial coefficients, which can be efficiently computed.
 * D.Lazard (talk) 11:28, 18 September 2021 (UTC)

Oh I dunno if there’s a formal assessment process, outside editors putting up their best guess. I was just going by my personal impression, but maybe I'm out of step with the criteria as applied to other math articles. My thoughts were: there is no detailed citation of sources (about half of the sections have no sources cited and no links to more detailed discussion; there have been thousands of research articles referencing the complex logarithm, surely there are more relevant to this article than what is currently cited), no discussion of the history, essentially no discussion of the applications inside or outside of mathematics, minimal discussion of computation, very limited figures. The word “cylinder” does not appear, though the complex logarithm is the conformal map from plane (or Riemann sphere) to cylinder and no discussion of the Mercator projection; the relation between the complex logarithm and trigonometry is barely discussed. The different sections read a bit choppy, and there are some differences of notation. The article does not seem at all friendly for non-experts, e.g. the domain excluding the origin is briefly mentioned in the lead section $$U \subseteq \mathbb{C}^\times := \mathbb{C} \setminus \{0\}$$ but this is never described in a way friendly for non-mathematicians, and the comment that "the origin a branch point of the function" only appears in a figure caption. I dunno, maybe B class is an okay description, but I think this article could be dramatically better. –jacobolus (t) 19:39, 18 September 2021 (UTC)
 * There is some discussion of analytic continuation already in the later sections of the article, but perhaps more could be added. The property of the complex logarithm as a holonomic function is not mentioned in most complex analysis textbooks, I think, and is not a property that usually springs to mind when mentioning the complex logarithm, but yes, it could be added as a remark.  Yes, history should be added.  Yes, some references to applications in engineering, say, could be added.  As for discussion of applications inside mathematics, it is a little like discussing applications of addition in mathematics - yes, it is used in many places, but what is particularly relevant to the topic at hand?  The article already says all one needs to know to compute the complex logarithm.  Yes, adding a few more figures could be useful, such as a simple one showing the complex plane with the standard branch cut along the negative real axis.  There were some other figures in older versions of the article, but they were removed because they had little mathematical content that was relevant.  The complex logarithm is not defined on the whole plane, let alone the Riemann sphere.  To me, Mercator projection is only tangentially related to complex logarithms; do any mapmakers actually use complex logarithms?  What more would one want to say about complex logarithms and trigonometry?  The differences in notation are intentional and used in the outside literature, because they reflect the different notions of complex logarithm; I think the usage in the article is currently consistent.  I agree with jacobolus that $$U \subseteq \mathbb{C}^\times := \mathbb{C} \setminus \{0\}$$ could be made more friendly, so I rewrote this to make it friendlier, as was suggested.  Yes, the fact that 0 is a branch point could be more prominent.  I'll restore the B class, but with the understanding that there are still several things to improve, as jacobolus pointed out. Ebony Jackson (talk) 22:26, 18 September 2021 (UTC)
 * Upon thinking some more, I agree with D.Lazard that it would be better to have a more detailed section (somewhere later in the article) devoted to analytic continuation. Also, jacobolus, when you were referring to complex logarithms and trigonometry, did you mean the expressions for inverse trigonometric functions in terms of the complex logarithm?  I agree that that is worth adding. Ebony Jackson (talk) 22:47, 18 September 2021 (UTC)
 * I added a mention of inverse trig functions. Should we add the actual formulas? Maybe let people go to inverse trigonometric functions for those? Ebony Jackson (talk) 23:12, 18 September 2021 (UTC)

The #1 most important thing to say about the logarithm (any kind), especially for a resource like Wikipedia intended to be read by a wide audience, is that it functions as a change of variables which converts multiplication in the domain to addition in the codomain. The current article doesn’t even really mention this explicitly, and the current lead section is much too technical and entirely unfriendly for a non-mathematician audience: it gets caught up in the minutiae, but misses the main idea.

Sometime within the first 2–3 sections after the lead, there should be a lay-accessible explanation of the geometric meaning of complex multiplication (rotation & scaling) vs. addition (translation), and some pictures showing how the logarithm converts one type of transformation into the other. It is crucially important to emphasize that the logarithm excludes the origin (for which multiplication collapses) and sends the multiplicative identity 1 to the additive identity 0. Geometrically/conceptually, the logarithm maps a plane minus the origin onto a two-infinite-ended cylinder. Turning the cylinder corresponds to rotating the plane but in cylindrical coordinates rotation is a kind of translation (addition); sliding the cylinder corresponds to scaling the plane. The ordinate measuring along a cylinder is unique but the ordinate measuring around the cylinder, like an angle measure for a point around a circle, is non-unique; as a result we can also think of the logarithm as a "multifunction" or as having a "principle branch": we can cut the cylinder lengthwise and stop associating the two sides, and then it becomes a two-infinite-ended strip.

Defining the logarithm in terms of the exponential function is done in mathematics textbooks for the sake of cutting down exposition and simplifying proofs (mathematics textbooks rely on ordering concepts in some linear order, and proving everything based on previous proofs), but shouldn’t be the only (or maybe even the primary) definition given here, and shouldn’t be belabored in the lead section. Instead a definition as the integral of $dz/z$ should be emphasized, because this emphasizes the relevance of scale invariance. Alternately the logarithm can be defined as the limit of a power function (geometrically, a map from the plane onto a cone) as the power approaches 0 (cone gets infinitely pointy): $\log z = \lim_{a \to 0} (z^a-1)/a.$ Arguably the logarithm is a more fundamental concept than the exponential (because the plane is the canonical domain for complex arithmetic, not the cylinder), and the exponential should properly be defined as its inverse; the main reason they are introduced and thought about in the opposite way is that it is convenient for the narrative of particular (dry and technical) textbooks.

The logarithm can also be described as a Schwarz–Christoffel mapping with one prevertex at the origin and the other at infinity. The function composition of the logarithm with a Cayley transform (e.g. $$z \mapsto 2 \operatorname{artanh} z$$) maps the unit disk onto the two-infinite-ended strip (or if you like, the plane minus two points onto the cylinder). This is useful e.g. for numerical integration of functions with singularities at one or both endpoints of an interval. This composition is also the basis of Bipolar coordinates in the plane (also see: Apollonian circles, pencil of circles, Radical axis, Inscribed angle). Logarithms are employed in making Schwarz-Christoffel maps from a 2-infinite-ended strip (instead of half-plane or disk) to arbitrary polygonal domains.

The complex logarithm is the very foundation of trigonometry: it is the function which converts rotations of the plane (which are naturally described as unit complex numbers) to (bivector-valued, a.k.a. "pure imaginary") angle measures. The logarithm more generally also maps from rotations to angle measures, (e.g. for 3D rotations from Euler parameters to the Axis–angle representation).

As for the Mercator projection, this is a connection that can e.g. be found in Felix Klein, Elementary Mathematics from an Advanced Standpoint (p. 103). It should probably be mentioned more prominently in the Wikipedia article about the projection, as it greatly simplifies many things. Since geometrically the complex logarithm and the Mercator projection (starting from the stereographic projection, see Riemann sphere) are one and the same, much of the history of the Mercator projection can be seen as a kind of pre-history of the complex logarithm. While we are at it the Gudermannian function should probably be mentioned here. To answer your question Ebony Jackson: Using complex variables is certainly something done by cartographers and geodesists (notice that many of the great mathematicians were also employed in cartography/geodesy, e.g. Euler, Gauss, Lagrange, Bessel, Chebyshev, ....)

How is it that this article doesn’t even link to Logarithmic spiral?

The logarithm gets used throughout mathematics because making a change of variables to convert products to sums or sums to products is often convenient; some section can discuss the times when one or another operation is easier to work with and show how logarithms are employed. For example, the logarithm converts Laurent polynomials to Trigonometric polynomials, something not currently mentioned in this article.

Etc. Etc. There is a huge amount of stuff that can be linked to complex logarithms, I am sure I am only scratching the surface in the above. A wiki article can try to do a whole lot more than the few pages of introduction found in an undergraduate textbook.

–jacobolus (t) 23:28, 18 September 2021 (UTC)
 * I’ve been hunting for sources that talk about the complex logarithm/exponential as mapping from plane <-> cylinder. Here are three:
 * –jacobolus (t) 20:36, 19 September 2021 (UTC)
 * Here is a paper applying the complex logarithm to analyzing the shape of spiral galaxies:
 * –jacobolus (t) 21:42, 19 September 2021 (UTC)
 * –jacobolus (t) 20:36, 19 September 2021 (UTC)
 * Here is a paper applying the complex logarithm to analyzing the shape of spiral galaxies:
 * –jacobolus (t) 21:42, 19 September 2021 (UTC)
 * –jacobolus (t) 21:42, 19 September 2021 (UTC)

logarithmic branch points is currently red links.--SilverMatsu (talk) 15:08, 19 September 2021 (UTC)
 * Could be added to Singularity (mathematics) perhaps. Mathworld has a page at http://mathworld.wolfram.com/LogarithmicSingularity.html –jacobolus (t) 17:53, 19 September 2021 (UTC)
 * The reason that the article doesn't open by saying that the complex logarithm converts multiplication into addition is that this is false for any branch of the complex logarithm - this is a very common error. (Yes, you can kind of make it true by considering multivalued functions, but this gets rather technical if you want to say what this means.)  Many one-variable calculus textbooks do define log before exp (in the context of real functions).  In modern science and engineering, exponential functions arise most commonly when the rate of change of a quantity is proportional to the quantity (the differential equation governing population growth, interest, cooling, radioactive decay, etc.), and it is only incidental that they convert addition to multiplication.  Second order constant-coefficient ODEs are the primary source of complex exponential functions in science and engineering.  Similarly (real) logarithms, although historically used to aid in multiplication, arise most commonly in applications as the inverse of an exponential function - e.g., to compute the doubling time of a population, or the half-life of an radioactive isotope.  To view log as a mapping from the punctured plane to the cylinder requires coordinatizing both in a particular way; otherwise you can use other functions.  Defining the complex logarithm by $\log z = \lim_{a \to 0} (z^a-1)/a$  is circular, since the usual definition of $$z^a$$ involves the complex logarithm.  Thank you for all the references, especially Klein. Ebony Jackson (talk) 01:57, 29 September 2021 (UTC)
 * It is not "false". It just requires a bit of care. For instance, we can say:
 * $$zw = \exp(\log z + \log w)$$
 * This is making a change of variables by taking the log to move from complex multiplication (scaling and rotation) in the plane to complex addition (translation) in the cylinder. It doesn’t matter which angular coordinate you choose in the cylinder, because we transform back to the plane afterward. (If you wanted you could build a slide rule in the form of two concentric cylinders with a transparent one on the outside, with which you could compute complex multiplication by sliding/rotating the outer cylinder relative to the inner.) It is certainly important to be precise about the details, but that doesn’t change the basic fact that the main idea/purpose of the logarithm is to convert multiplication into addition. Leaving that out is missing the forest for the trees. “make it true by considering multivalued functions” – Nah, the problem is that the way we assign coordinates is by unrolling the cylinder onto a plane. The “multi-valued function” is straight-forwardly single-valued on the cylinder; it just happens that there are multiple ways of describing points on a cylinder using unrolled-into-the-plane coordinates. “exponential functions arise most commonly when the rate of change of a quantity is proportional to the quantity” – Precisely right: logarithmic space arises naturally when we examine changes that are multiplicative (at each time step) over linearly proceeding time, so that we want to treat multiplication additively! “Defining the complex logarithm [as the limit of power functions] is circular” – That depends entirely how you define your power functions. For example you can define complex nth roots using nothing but high school trigonometry if you want to, and then take the complex logarithm to be $\log z = \lim_{n \to \infty} n\bigl(\sqrt[n]z-1\bigr)$ . Note I am not suggesting this should be people’s primary definition, but only that this is a different point of view that helps elucidate the concept and its geometry. [Aside: "complex logs are circular" is a nice pun]. –jacobolus (t) 03:04, 29 September 2021 (UTC)
 * This is an interesting way of considering logarithms. However it is unclear how to make this understandable by a wide audience. In particular, one may not suppose that readers know geometry (cylinders), continuous mappings from planes to cylinders, and the relationship of these geometrical concepts with complex analysis. This makes problematic to follow your suggestions. D.Lazard (talk) 11:09, 29 September 2021 (UTC)
 * If we are worried about what prospective readers might know, the current version more or less assumes they are an upper division undergraduate pure math student (or beyond) who has already taken a complex analysis course. I think we could do a lot more to make this article accessible to laypeople, high school students, etc. But sure I agree that a clear geometrical explanation takes considerable effort with lots of pictures, etc. I’ll try to give it a shot if I get a chance, but no promises about when that might be. –jacobolus (t) 23:48, 29 September 2021 (UTC)
 * The article must be accessible to a wider public than students of a given level, and this public may be not interested in geometric aspects. For example, programmers who want implement the complex logarithm function, and electrical engineers. This has to be taken into account in every major edit. D.Lazard (talk) 08:16, 30 September 2021 (UTC)
 * “ must be accessible to a wider public” – it is currently almost entirely inaccessible to a “wider public” (i.e. anyone without extensive undergraduate-level math education). Someone who went through a rigorous engineering program including a complex analysis course should be fine, but the median programmer (or any other layperson) is going to be challenged by the pre-requisite assumptions, lack of context, lack of diagrams, and lack of non-technical or semi-technical explanation. –jacobolus (t) 18:36, 11 October 2021 (UTC)
 * A lot of math articles are way worse off than this one. It's hard for people with a rigorous understanding of an abstract topic to appreciate what kind of help or diagram might be most useful to those not already in the know.  So let's invite improvements, but not be too critical of what's there.  But surely most math and engineering undergraduates would know enough about complex numbers and logarithms to read the article and learn about complex logarithms; sadly, not so true with computer scientists/programmers.  Dicklyon (talk) 18:59, 11 October 2021 (UTC)
 * By the way, in my book I have a "Complex Logarithm History" box that has a small slice of the history, with Roger Cotes c. 1740, and Euler, about the log of a unit-magnitude complex value (a pure imaginary logarithm), from which the rest follows by adding the real log of the magnitude. Dicklyon (talk) 19:06, 11 October 2021 (UTC)

The captions for the spiral range of log should contain the note that the drawing only shows data for z outside the unit circle. Since ln(1) =0, all point on the unit circle are mapped to the axis. Log(z) for z inside the unit circle are not shown. — Preceding unsigned comment added by SemiSalt (talk • contribs) 16:03, 24 April 2024 (UTC)

Complex multiplication slide rules
I’m moving this to a new section. Hope you don’t mind Dick. –jacobolus (t) 06:19, 9 October 2021 (UTC)


 * Re that slide rule, it's done with one cylinder and two cursors. I had one, but I sold it, and don't seem to have retained a photo.  The complex logarithm certainly does work to reduce complex multiplication to 2D addition. Dicklyon (talk) 19:07, 8 October 2021 (UTC)

Cool! Wikipedia has Fuller calculator and a web search turned up:
 * “Identifying all complex numbers that differ by a multiple of $2π$, we obtain an infinite cylinder as the quotient space, and the logarithm can be considered as a single-valued function from the non-zero complex numbers onto this cylinder.”
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * “Identifying all complex numbers that differ by a multiple of ᙭᙭᙭, we obtain an infinite cylinder as the quotient space, and the logarithm can be considered as a single-valued function from the non-zero complex numbers onto this cylinder.”
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg
 * Picture of David Whythe, https://www.sliderulemuseum.com/Ephemera/David_Whythe_UKSRC_1980_InventorOfWhytheComplexNumberSR.jpg

–jacobolus (t) 21:01, 8 October 2021 (UTC)