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Topic: Dedekind eta function

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	ONLINE THESES IN NUMBER THEORY
		Evaluation of the Dedekind Eta Function, PhD thesis, William Hart, Macquarie University 2004
		The pair correlation of the zeros of the Riemann zeta function, Senior Thesis, Lillian Pierce, Princeton University, 2002
		On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD thesis, Christopher Hughes, University of Bristol 2001
	www.numbertheory.org /ntw/N5.html (3211 words)

	PlanetMath: modular discriminant
		The Dedekind eta function is defined to be
		The Dedekind eta function should not be confused with the Weierstrass eta function,
		-invariant, as a function of lattices, is defined to be:
	www.planetmath.org /encyclopedia/DedekindEtaFunction.html (104 words)

	PlanetMath: modular discriminant
		The Dedekind eta function is defined to be
		The Dedekind eta function should not be confused with the Weierstrass eta function,
		-invariant, as a function of lattices, is defined to be:
	planetmath.org /encyclopedia/DeltaFunction2.html (105 words)

	Dedekind eta function - Wikipedia, the free encyclopedia (Site not responding. Last check: )
		The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.
		Because of these functional equations the eta function is a modular form of weight 1/2 and level 1 for a certain character of order 24 of the metaplectic double cover of the modular group, and can be used to define other modular forms.
		Because the eta function is easy to compute numerically from either power series, it is often helpful in computation to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
	www.sciencedaily.com /encyclopedia/dedekind_eta_function (407 words)

	PlanetMath: Weierstrass sigma function
		Note that the Weierstrass zeta function is basically the derivative of the logarithm of the sigma function.
		The Weierstrass eta function must not be confused with the Dedekind eta function.
		This is version 1 of Weierstrass sigma function, born on 2003-08-25.
	planetmath.org /encyclopedia/EtaFunction.html (117 words)

	Dedekind eta function - Definition, explanation
		The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive.
		The eta function is holomorphic on the upper half plane but cannot be continued analytically beyond it.
		Because the eta function is easy to compute, it is often helpful to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
	www.calsky.com /lexikon/en/txt/d/de/dedekind_eta_function.php (268 words)

	The Mandelbrot Set as a Modular Form
		Modular forms are a particular kind of function on the complex upper half-plane studied in analytic number theory and the theory of elliptic curves.
		It is closely related to the Dedekind eta function, which is a modular form.
		Such a function is provided by not working with the modulus, but subtracting the divergence directly; this is explored in the next section.
	www.linas.org /math/dedekind/dedekind.html (3929 words)

	Elliptic function Summary
		Although elliptic functions were simple in form, defined as r(x, p(x))dx where r(x,y) is a rational function in two variables and p(x) is a 3rd or 4th degree polynomial without repeated roots, the development of elliptical functions had profound consequences on the analysis of the mechanics of motion.
		In addition, Le Verrier's use of elliptical functions to describe a discrepancy in the orbital motion of Mercury (e.g., the advance of the perihelion of Mercury) became an important stimulus to the subsequent formation and proof of Albert Einstein's general theory of relativity.
		Historically, elliptic functions were discovered as inverse functions of elliptic integrals; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives.
	www.bookrags.com /Elliptic_function (2241 words)

	Dedekind eta function - Term Explanation on IndexSuche.Com (Site not responding. Last check: )
		is a function defined on the upper_half_plane of complex numbers whose imaginary part is positive.
		For any such complex number z, we may set ''q'' = ''e''2π''zi'', and define the eta function by :\et(az) = q^{1/24} \prod_{n=1}^{\infty} (1-q^n) The eta function is holomorphic on the upper half plane but cannot be continued analytically beyond it.
		The eta function satisfies the functional equations :\et(az+1) = \exp(2 \pi i/24)\eta(z) :\eta(-1/z) = \sqrt {-iz} \eta(z) Because of these functional equations the eta function is a modular_form of weight 1/2, and can be used to define other modular_forms.
	www.indexsuche.com /Dedekind_eta_function.html (210 words)

	List of mathematical functions - Wikinfo
		A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.
		Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena.
		Ackermann function: in the theory of computation, a recursive function that is not primitive recursive.
	www.wikinfo.org /wiki.php?title=List_of_mathematical_functions (1513 words)

	Approximations to via the Dedekind eta function (Site not responding. Last check: )
		This connection is shown utilizing their {\it alpha}-function which is defined in terms of theta-functions.
		They are able to find $p$-th order iterations based on this function using modular equations for the theta-functions.
		In this paper we construct an infinite family of functions $\alpha_p$.
	www.cecm.sfu.ca /organics/papers/garvan/index.html (197 words)

	Elliptic and Modular Functions
		More information on elliptic functions can be found for example in Chandrasekharan [Cha85], and for modular functions and their use see Koblitz [Kob84].
		Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the normalized q-series expansion of the discriminant Delta(q) evaluated at tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.umich.edu /~gpcc/scs/magma/text570.htm (1235 words)

	Catalogue of GP/PARI Functions: Transcendental functions
		U-confluent hypergeometric function with parameters a and b.
		This function is analytic on the complex plane with non-positive integers removed.
		Note that this function is also called whenever one multiplies two objects which are known to be identical, e.g.they are the value of the same variable, or we are computing a power.
	pari.math.u-bordeaux.fr /dochtml/html.stable/Transcendental_functions.html (1982 words)

	Release Notes V2.4 (December 14, 1998)
		The function TransitiveGroupIdentification(G) returns the index in the transitive group database of the group that is isomorphic to a given transitive group G. A recursive algorithm due to Bill Kantor for computing the maximal normal p-subgroup of a permutation group is now available as the function pCoreKantor.
		New function VectorSpace(G) to return the vector space on which the matrix group G acts naturally (assuming G is over a field; otherwise the same as the function RSpace).
		The function ClassRepresentative, applied to an ideal I belonging to an order whose class group is known, will give the chosen representative of the ideal class containing I. The group of S-units corresponding to a set S of prime ideals is provided by the function SUnitGroup.
	www.math.colostate.edu /manuals/magma/htmlhelp/text71.html (5496 words)

	Elliptic and Modular Functions
		More information on elliptic functions can be found for example in Chandrasekharan [Cha85], and for modular functions and their use see Koblitz [Kob84].
		Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the q-series expansion of the discriminant Delta(q) evaluated at q = e^(2pi itau) where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.math.lsu.edu /magma/text582.htm (1211 words)

	The Mandelbrot Set as a Modular Form
		Modular forms are a particular kind of function on the complex upper half-plane studied in analytic number theory and the theory of elliptic curves.
		This mapping is curious because it is not infrequent in the literature, and because a periodic function on the upper half-plane takes the appearance of a self-similar function on the disk.
		Such a function is provided by not working with the modulus, but subtracting the divergence directly; this is explored in the next section.
	linas.org /math/dedekind/dedekind.html (3929 words)

	Mathematical Research Letters :: a mathematics journal
		A fundamental formula in the study of elliptic functions is the product formula for Dedekind's eta function or, equivalently, for the holomorphic cusp form on the upper half plane $\bold h$ which is of weight $12$ with respect to the action by $PSL(2, \bold Z)$.
		A related formula expresses the determinant of the Laplacian which acts on the space of smooth functions on an elliptic curve with a period of the elliptic curve and the Dedekind eta function.
		In \cite{JT 94a}, we constructed a holomorphic function on the moduli space of marked, polarized, algebraic $K3$ surfaces of fixed degree using determinants of Laplacians.
	www.mrlonline.org /mrl/1995-002-003/1995-002-003-013.html (174 words)

	The On-Line Encyclopedia of Integer Sequences
		Also the q-expansion of the Dedekind eta function without the q^(1/24) factor.
		B. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer.
		G.f.: q^(-1/24)eta(z), where q=exp(2 Pi i z) and eta* is the Dedekind eta function.
	www.research.att.com /~njas/sequences/A010815 (604 words)

	Publications of William Hart
		Evaluation of the Dedekind Eta Function - with Robin Chapman - Canad.
		An Identity for the Eta Function Involving Two Independent Complex Variables (with Bruce Berndt - to appear in the Bulletin of the London Mathematical Society)
		Eta Evaluations from Modular Equations for Weber Functions
	www.maths.warwick.ac.uk /~masfaw/preprint.html (410 words)

	Springer Online Reference Works (Site not responding. Last check: )
		Dedekind [a1] showed that this quantity occurs in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group.
		Several elementary proofs of this relation can be found in [a2].
		These proofs exhibit other interpretations of Dedekind sums, related to counting lattice points and Fourier theory (cf.
	eom.springer.de /d/d110070.htm (186 words)

	CMB - Evaluation of the Dedekind Eta Function
		CMB - Evaluation of the Dedekind Eta Function
		We extend the methods of Van der Poorten and Chapman for explicitly evaluating the Dedekind eta function at quadratic irrationalities.
		Further, we overcome the limitations of the earlier methods and via modular equations provide explicit evaluations where the class number is 5 or 7.
	journals.cms.math.ca /cgi-bin/vault/view/chapman8529 (84 words)

	Andrey Todorov
		The generalization of the Dedekind Eta Function to higher dimension, its relation with the regularized determinants of CY metrics and with the algebraic notion of the discriminants.
		Bass and A. Todorov: The analogue of Dedekind eta function for CY manifolds I.
		Jorgenson and A. Todorov: A Conjectured Analogue of the Dedekind Eta Function for K3 Surfaces.
	www.math.ucsc.edu /Faculty/Todorov.html (404 words)

	Math. Phys. seminars (Site not responding. Last check: )
		We employ results on the effects of modular transformations on the eta function to simplify his formula.
		Suppose the cylindrical spatial domain is viewed as a rectangle equipped with periodic boundary conditions in one direction, and let the rectangle be mapped conformally onto the upper half of the complex plane, with its four vertices mapped to points on the real line.
		I will present a calculation of the partition function for this quantum field theory which is conjectured to be the continuum limit of the XY-model at low temperatures.
	math.arizona.edu /~mathphys/s04_seminars.html (1531 words)

	resplan (Site not responding. Last check: )
		For my dissertation, I am formalizing this general rule and applying the results to the study of known modular forms (and related arithmetic functions) as well as the discovery of new modular forms.
		The Dedekind eta function and the partition function.
		to be a log-polynomial function, or a rational function, or (in the most general case) an element of
	www.math.temple.edu /~ludwick/resplan/resplan.html (918 words)

	The Analogue of the Dedekind Eta Function for CY Manifolds., , , November 27, 2006
		The Analogue of the Dedekind Eta Function for CY Manifolds.,,, November 27, 2006
		Â is a bounded function on the moduli space of the elliptic curves.
		It is a classical fact that the Dedekind eta function is related to the algebraic discriminant of the equation that defines the elliptic curve.
	hermes.aei.mpg.de /arxiv/05/03/231/article.xhtml (1896 words)

	Elliptic Lambda Function
		-modular function defined on the upper half-plane by
		are known as elliptic integral singular values, and can be expressed in terms of a finite number of gamma functions (Selberg and Chowla 1967).
		Dedekind Eta Function, Elliptic Alpha Function, Elliptic Integral of the First Kind, Elliptic Modulus, Elliptic Integral Singular Value, j-Function, Jacobi Theta Functions, Klein's Absolute Invariant, Modular Function, Modular Group Lambda, Ramanujan g- and G-Functions
	users.skynet.be /fa956617/math/topics/EllipticLambdaFunction.html (219 words)

	Twelve @ iCookClub.com
		Twelve is the smallest weight for which a cusp form exists.
		This cusp form is the discriminant Î”(q) whose Fourier coefficients are given by the Ramanujan Ï-function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function.
		This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function function at -1 i.e.
	reference.icookclub.com /twelve.html (575 words)

	week126
		Besides the Riemann zeta function, there are a lot of other special functions that show up in the study of elliptic curves.
		This was first worked out by Planck, who assumed the harmonic oscillator had discrete, evenly spaced energy levels and computed its partition function as part of his struggle to understand the thermodynamics of the electromagnetic field.
		To get the partition function of a quantum system built from a bunch of noninteracting parts, you multiply the partition functions of the parts (since the trace of a tensor product of operators is the product of their traces).
	math.ucr.edu /home/baez/week126.html (2516 words)

	Catalogue of GP/PARI Functions: Transcendental functions
		Euler I Pi ^ abs acos acosh agm arg asin asinh atan atanh bernfrac bernreal bernvec besselh1 besselh2 besseli besselj besseljh besselk besseln cos cosh cotan dilog eint1 erfc eta exp gamma gammah hyperu incgam incgamc lngamma log polylog psi sin sinh sqr sqrt sqrtn tan tanh teichmuller theta thetanullk weber zeta
		If the exponent is an integer, then exact operations are performed using binary (left-shift) powering techniques.
		In the present version 2.3.1, this function is not very accurate when x is small.
	wotan.algebra.math.uni-siegen.de /~countnumber/parigp-ref/Transcendental_functions.html (2034 words)

	Amazon.com: Modular Functions and Dirichlet Series in Number Theory (Graduate Texts in Mathematics): Books: T.M. Apostol (Site not responding. Last check: )
		Most of this book is devoted to a classical treatment of elliptic and modular functions with some of their number-theoretic applications.
		Among the major topics covered are Rademacher's convergent series for the partition modular function, Lehner's congruences for the Fourier coefficients of the modular function j, and Hecke's theory of entire forms with multiplicative Fourier coefficients.
		It is an excellent intro to modular forms, modular functions, the j-invariant, the Weierstrass elliptic functions and the Hecke operators, in the context of the modular group SL(2,Z).
	www.amazon.com /Modular-Functions-Dirichlet-Graduate-Mathematics/dp/3540971270 (1220 words)

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