Talk:Theta function

Application

Is there some canonical use for this that I simply can't see? Perhaps in physics or some `simple' math. 19:28, 12 Feb 2005 User:Ub3rm4th

OK, I added a section on the heat equation and also the Heisenberg group. Does that work for you? Its also studied in quantum field theory specifically D-branes and string theory.linas 04:21, 2 Mar 2005 (UTC)

Not sure if one should add this, but an application in 'simple' math is that various expressions in the values of \theta(z,\tau), for fixed \tau and for z varying, parametrize an elliptic curve if it is viewed as the solution set of a cubic equation. So if someone gives you a cubic polynomial equation in 2 variables and says parametrize the solution set, you can say it is parametrized by the complex number z modulo integers and integer multiples of \tau, and particular expressions are the parametrization. A difficulty is that people who read the article are likely already to know this. It might be better to find a reference to an elliptic curve article somehow.Createangelos (talk) 00:40, 9 November 2013 (UTC)

I agree that Wikipages on mathematical functions should give some discussion on applications. Why were these functions invented or identified in the first place? Also useful to know. With respect to theta functions, for me, I've encountered theta functions when inverse Laplace transforming coth[sqrt[s]]/sqrt[s]. Spanier and Oldham (An Atlas of Functions, 1987, 27:13) note that theta functions often arise in the context of Laplace transforms. So, this might translate into a more general application for problems of impulse response. Sincerely, DoctorTerrella (talk) 17:06, 3 September 2014 (UTC)

theta function gives green's function for heat equation ? i dont get the proof.

In Mumford's paper they say theta function gives fundamental solution to the heat equation. To show that i miss the differential operator of the heat equation applied to the theta-function (distribution) for t=0. You say (and Mumford shows) that lim{t->0} theta(x, it) = delta(x) . but what does that help for showing its a fundamental solution?? (unsigned anonymous post 1 jan 2006)

I don't understand the question being asked. Can you rephrase? linas 23:53, 3 January 2006 (UTC)

i think it must be shown: Heat(theta(x, it)){t->0} = delta(x)delta(t) in order to show that theta(x, it) is a fundamental solution, where Heat() shell be the differential operator of the heat equation. (unsigned anonymous post 5 jan 2006)

I still don't understand what you are saying or asking. It is relatively straightforward to to demonstrate that the theta is a solution to the heat equation, and that it satisfies the periodic delta function boundary condition. Are you saying that you are unable to derive this proof on your own? Wikipedia is not the place for long, detailed proofs, which is why this article doesn't have one. If you need help with an equation, you might try asking a question at Wikipedia:Reference desk/Mathematics. linas 23:33, 5 January 2006 (UTC)

PlanetMath incursions

Someone has made a hash of this article by dumping unedited stuff from PlanetMath in here. Since the notations differ, this was not a good idea. I think I might reedit it to conform, and remove the PlanetMath tags. Gene Ward Smith 22:07, 6 June 2006 (UTC)

I've got the notation consistent now. Gene Ward Smith 05:49, 7 June 2006 (UTC)

Other notations

The variety of notations for theta functions must be one of the nightmares of applied maths as others have noted! The existence of this section and a separate Jacobi_theta_functions_(notational_variations) wikipedia page is confusing. Neither seems to have been edited for some time, which makes me think people are using a non-Wikipedia source for disambiguation. From my limited use I think the transformations of the nome should be emphasised as early as possible. The identities of the infinite sums, Sum_{k=- infinity}^{infinity} q^((k+a)^2), where a is 0, 1/2 or general and with (-1)^k, were the way I found my way into the theta functions and I would think they were fairly important. I would call these "nome in both arguments" as opposed to "nome" in one argument, but have no idea if that is standard. I would include mention of Neville's notation, as per A&S also. Nick Mulgan (talk) 01:09, 11 March 2017 (UTC)

How about some mention of, or better yet, a separate article on, the lack of standardization in notation? Abramowitz and Stegun has theta sub four; Whittaker and Watson include a table which shows four other notations not shown here, and some of these define the functions with period pi instead of 1.Cstaffa 02:33, 13 February 2007 (UTC)

I've thought about this a little bit, Cstaffa, but have not yet reached a conclusion about the best course of action. This article does mention Jacobi's original notation already, although it doesn't say anything about the transformation of variables that leads to period/quasi-period of either (1, τ), or (π, πτ).

Oh – it's not just period π. Two of the functions [the ones involving fractional powers of q, or cos(2n+1)z / sin(2n+1)z] have period 2π, just like the trigonometric functions. But a lot of authors, including Whittaker and Watson, gloss over this point.

I think that A&S implicitly adopted the notation of Tannery & Molk (they refer to W&W, who acknowledge the French guys T&M for the version in Modern Analysis). Anyway, I'm curious … what purpose do you think it would serve to discuss all the different notations for the Theta functions on Wikipedia? Perhaps if we identify a reason why, we can help each other understand the best way to go about it. DavidCBryant 03:09, 13 February 2007 (UTC)

It would help mopes like me who don't usually deal with such functions who are trying to use formulae pulled from references. I decided to compute g2 for the Weierstrass function with periods 1/2 and 1/2i, so as to get the argument for the parameterization of Costa's minimal surface. I was using A&S and then checked Wikipedia. Not having seen W&W yet, I was confused by the difference in notations. This unsigned comment is from Cstaffa 16:26, 13 February 2007 (UTC)

Here, don't be too hard on yourself. I had to go look up "Costa's minimal surface" because topology isn't my bag. :-) And I think the various systems of notation are confusing, also. (I don't particularly like the double-subscript notation this article uses, but I didn't write the article – I just stumbled across it and decided it needed some cleanup.)

I can't promise how soon I'll get it done, but I'll put W&W's little table in an article somewhere, and link to that article from this one. Where do you think the link ought to go, Cstaffa? I'm thinking a short italicized sentence at the very beginning to the effect of "See x article for different systems of notation used with Theta functions" would probably work best for someone coming at it from your angle. DavidCBryant 18:29, 13 February 2007 (UTC)

I think that sounds ideal. I'll have a go at starting it if you don't get to it first.Cstaffa 18:43, 13 February 2007 (UTC)

I am undertaking to fix the following problems:

The first Jacobi theta function presented is given as θ(z;τ). It should be θsub00. The notations are confusing enough without introducing a novel one. The function shown does not correspond to Jacobi's θ, as is shown on page 487 of Whittaker and Watson.

Further down, θsub01 is identified with Jacobi's θsub0. According to the above reference, Jacobi did not have a θsub0, but rather a bare θ.

Under Theta functions in terms of the nome, the notation θ(z;q) is used, which conflicts with the usage established previously of θ(z;τ) and θ(z,q). Further along this is given as θ(z|q). Cstaffa 19:58, 14 February 2007 (UTC)

Someone please check Mumford to see if this follows his notation. I don't have easy access to Mumford. Cstaffa 20:53, 14 February 2007 (UTC)

suspicious edit

I don't know anything about theta functions, but this edit might be suspicious. The identity before the identity is on Mathworld anyways. Akriasas 06:18, 27 February 2007 (UTC)

Good catch, Akriasas. Thank you. DavidCBryant 12:30, 27 February 2007 (UTC)

Transformation under $\tau \mapsto \tau +1$

In the section on Jacobi identities, it says:

Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ+1 and τ ↦ -1/τ. We already have equations for the first transformation;

However, while it has certainly been mentioned that the theta function is periodic of period $1$ in $z$ for $\tau$ fixed, I can't find any mention of the transformation $\tau \mapsto \tau +1$. Am I missing it? JadeNB (talk) 02:50, 15 May 2008 (UTC)

Hi,

I noticed this too, the equations for τ ↦ τ+1 aren't given anywhere in the article. This worried me a little, but they are easy to work out since adding 1 to $\tau$ in $e^{i\pi n^{2}\tau +2i\pi nz}$ has the same effect as adding 1/2 to z -- becuase $n$ is congruent to $n^{2}$ modulo 2. I added this comment just now, though it is a little irrelevant to the section title.Createangelos (talk) 11:34, 12 November 2013 (UTC)

Incorrect explicit values

Some of the explicit values shown are incorrect. theta(2) is roughly 1+2exp(-2pi)=1.00 and theta(1) is roughly 1 + 2exp(-pi)=1.08. So the ratio is 0.92. But the ratio of the explicit values expressions is (2+sqrt(2))^0.25/2 = 0.68. I'm guessing theta(2) is wrong. theta(3)/theta(1) and theta(4)/theta(1) both yield a value around 0.92 which suggests they are correct. theta(5)/theta(1) is too small by a factor of 25. —Preceding unsigned comment added by 209.67.107.10 (talk • contribs) 21:27, 30 July 2009

graphed on a polar coordinate system

I removed the sentence A theta function is graphed on a polar coordinate system from the introduction. It seems wrong: at least, any function can be graphed in any coordinate system you like. It is not amplified in the text. Deltahedron (talk) 16:51, 9 May 2012 (UTC)

The name of the article

Wouldn't it be better if the article had the name "Jacobi theta functions"? The article is mentions generalizations in a section, but everything else is about Jacobi theta functions. K9re11 (talk) 22:44, 10 March 2015 (UTC)

Line bundle

It would be nice to elaborate on (or at least include a reference for) that last sentence about line bundles and descent; it is mentioned once and then never again in the article. Michael Lee Baker (talk) 19:25, 6 June 2015 (UTC)

Graphs of theta functions

I may be interpreting something wrong here, but the 3D graphs of the four theta functions in this article don't specify whether it's the modulus, the real part or the imaginary part of the theta function that is the z-coordinate in the surface plot.

Isojarv1 (talk) 18:09, 10 July 2019 (UTC)

Character differences in notation

Some of the expressions in this article use the cursive theta (ϑ or $\vartheta$) and others use the regular lowercase theta (θ or $\theta$). Is the distinction meaningful, like the distinction between ℒ and L and L? If they are different, it would be nice to explain the different meanings, but if they are the same, it would be nice to make the typography consistent to avoid confusing readers just starting to learn the concept. -- Beland (talk) 21:17, 10 August 2023 (UTC)

@Beland: See the section Theta function#Auxiliary functions. --JBL (talk) 17:46, 11 August 2023 (UTC)

@JayBeeEll: I read through that section, looked at the rest of the article carefully, and I'm still left with questions. I think the pattern is that ϑ(z;τ) and theta with double-number-subscripts are written with the cursive theta, and theta with a single-number-subscript is written with a regular theta? But is that a somewhat random choice made by this article or is it a standard convention? Is the distinction meaningful, or merely stylistic? For example, if I'm hand-writing these equations, do I need to carefully distinguish between cursive and regular theta, or can I just write them all the same? I see that Jacobi theta functions (notational variations) uses cursive theta with single-number-subscript. Is that an error, an arbitrary choice, or a different system of notation? -- Beland (talk) 17:57, 11 August 2023 (UTC)

Incomprehensible?

Regarding:

Die mentioned up directed Delta triangle...

It seems incomprehensible. E.g., is "Die" the German word for "The" ("the")?

--Mortense (talk) 19:44, 2 April 2024 (UTC)