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Topic: Theta function

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Elliptic function

Gamma function

Beta function

Riemann zeta function

Exponential function

Schwarzian derivative

Cousin problems

Method of steepest descent

The analytization trick

Moduli space

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	PlanetMath: functional equation for the theta function
		The lemma used in the derivation of the functional equation for the Riemann Xi function.
		This functional equation is not as remarkable as the one for the Xi function, because it does not actually extend the domain of the function.
		This is version 7 of functional equation for the theta function, born on 2003-01-29, modified 2006-11-25.
	www.planetmath.org /encyclopedia/FunctionalEquationForTheRiemannThetaFunction.html (166 words)

	NationMaster - Encyclopedia: Theta function (Site not responding. Last check: 2007-10-19)
		In mathematics, the Riemann-Siegel theta function is the function
		With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems).
		This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group.
	www.nationmaster.com /encyclopedia/Theta_function (716 words)

	Riemann-Siegel theta function - Wikipedia, the free encyclopedia
		In mathematics, the Riemann-Siegel theta function is the function
		The interest in the Riemann-Siegel theta function is in studying the Riemann zeta function and defining the Z function.
		For values with imaginary part between -1 and 1, the arctangent function is holomorphic, and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between -1/2 and 1/2, leading to a holomorphic function on this domain.
	en.wikipedia.org /wiki/Riemann-Siegel_theta_function (321 words)

	The relationship between foot function and theta angular correction
		Theta relates directly with the ability of any device, custom orthotic, arch supporting/foot controlling shoe modification or insert, constructed from any material, by any means of fitting or fabrication, to change foot function during weight-bearing use.
		By definition, theta is measured in the central 50 percent of that curve.
		Theta is the physical angle of support applied to the foot during function by any arch support or orthotic device.
	www.theta-orthotics.com /theta.html (629 words)

	Theta function - Wikipedia, the free encyclopedia
		The most common form of theta function is that occurring in the theory of elliptic functions.
		With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems).
		This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group.
	en.wikipedia.org /wiki/Theta_function (1143 words)

	PlanetMath: proof of functional equation for the theta function
		PlanetMath: proof of functional equation for the theta function
		"proof of functional equation for the theta function" is owned by Mathprof.
		This is version 11 of proof of functional equation for the theta function, born on 2003-02-11, modified 2006-10-13.
	www.planetmath.org /encyclopedia/ProofOfFunctionalEquationForTheRiemannThetaFunction.html (201 words)

	Brain wave Functions Alpha Beta Theta Brainwaves
		It is well known that the brain is an electrochemical organ; researchers have speculated that a fully functioning brain can generate as much as 10 watts of electrical power.
		A person who is driving on a freeway, and discovers that they can't recall the last five miles, is often in a theta state--induced by the process of freeway driving.
		During this awakening cycle it is possible for individuals to stay in the theta state for an extended period of say, five to 15 minutes--which would allow them to have a free flow of ideas about yesterday's events or to contemplate the activities of the forthcoming day.
	brain.web-us.com /brainwavesfunction.htm (1028 words)

	IRAF help : images gauss (Jan91)
		Theta is measured counter-clockwise from the x axis and must be between 0 and 180 degrees.
		The distance along the major axis of the Gaussian function at which the kernel is truncated in sigma pixels.
		Convolve an image with an elliptical Gaussian function whose sigma in the major and minor axis direction is 2.0 and 1.5 respectively, and whose position angle is 45 degrees, using wrap around boundary extension.
	www.astro.uni-bonn.de /~webstw/cm/lfa_html/iraf/images.gauss.html (704 words)

	Theta Functions (Site not responding. Last check: 2007-10-19)
		The vector c is called the characteristic of the theta function.
		This computes the multidimensional theta function with characteristic char (a 2g x 1 matrix) at z (a g x 1 matrix) and tau (a symmetric g x g matrix with positive definite imaginary part).
		This computes the multidimensional theta function with characteristic char (a 2g x 1 matrix) at z (a g x 1 matrix) and tau, the small period matrix of the analytic Jacobian A.
	magma.maths.usyd.edu.au /magma/htmlhelp/text583.htm (197 words)

	Beta Theta Pi Fraternity - Home
		Did you know that Beta Theta Pi has more than 750 alumni and Friends of Beta leading its undergraduate chapters to new heights?
		We've teamed up with officially licensed vendors to bring you unprecedented access to high-quality custom products for your next chapter function or event.
		Just visit www.greekquote.com for low prices and to support Beta Theta Pi's licensed vendor program.
	www.betathetapi.org (123 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		% MOMENTS % % Function calculates the moments of a binary image and returns % the centroid, the angle of axis of minimum inertia, and a measure % of 'roundness'.
		The function assumes that there is only one object % in the binary image.
		% % function [centroid, theta, roundness] = moments(im) % % Argument: im - a binary image containing values of 0 or 1 % % Returns: centroid - a 2 element vector % theta - the angle of axis of minimum inertia (radians) % roundness - ratio of minimum inertia/maximum inertia.
	www.csse.uwa.edu.au /~wongt/Documents/matlab/moments.m (137 words)

	CLAS Cerenkov detector efficiency
		Theta and Phy are accordingly polar and azimuthal angles of the entrance point on the surface of the Cerenkov detector in the sector reference system.
		The plot of mean number of photoelectrons as a function of projective angles Theta, Phy for sector 1 were used to plot the number of electron candidates as a function of Nphe.
		The possible decision could be to use Theta and Phy not for entring point, but for EC plane point, where we saw the self-focusing effect and used it for eid0 programm.
	www.jlab.org /~vlassov/cc/e_rec.html (908 words)

	PlanetMath: Riemann $þeta$-function
		The Riemann theta function is a number theoretic function which is only really used in the derivation of the functional equation for the Riemann Xi Function.
		Riemann showed that the theta function satisfied a functional equation, which was the key step in the proof of the analytic continuation for the Riemann Xi Function; this has direct consequences of course with the Riemann Zeta function.
		Cross-references: Riemann zeta function, Riemann Xi function, analytic continuation, functional equation, exact form, domain, functional equation for the Riemann Xi function, derivation, function
	planetmath.org /encyclopedia/RiemannThetaFunction.html (128 words)

	Theta function (Site not responding. Last check: 2007-10-19)
		The Jacobi theta function is a function defined for two complex variables z and, where z can be any complex number and is confined to the upper half plane, which means it has positive imaginary part.
		The Jacobi theta function can be thought of as belonging to a representation of the Heisenberg group in quantum mechanics, sometimes caled the theta representation.
		Then we see that the Jacobi theta function is an entire function of z that is invariant (mathematics) under G, and it can be shown that the Jacobi theta is the unique such function.
	read-and-go.hopto.org /Elliptic-functions/Theta-function.html (1204 words)

	CSE 250B Lecture Notes for May 22, 2000 (Site not responding. Last check: 2007-10-19)
		The M-step of EM is to maximize Q. The theorem says that for a fixed theta', maximizing Q gives the biggest possible increase in L. But the theorem does not say that L is maximized in one step.
		X, theta' ] where the expectation is an integral over Y. Often we can simplify this by finding a single special y value such that the integral over Y is the same as evaluating the integrand for this special y value.
		In general, the expectation of a function is the function of the expectation if and only if the function is linear.
	www-cse.ucsd.edu /~elkan/250B/may22.html (758 words)

	Elliptic and Modular Functions
		Defined this way, theta satisfies theta(q, - z)= - theta(q, z), it is periodic with period 2pi in the second variable: theta(q, z + 2pi)= theta(q, z), and its zeroes are of the form m_1pi + m_2(log x/i) for any integers m_1, m_2.
		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.math.wisc.edu /help/magma/text467.html (1285 words)

	CSE 291 Lecture Notes, January 20, 2004
		To show that a family is exponential, you may have to write theta as a function of the natural parameters, e.g.
		Generally, given x we want to guess theta such that p(x,theta) is high and d p(x,theta) / d theta = 0, to be at a local maximum for p(x,theta).
		Hence for fixed x, the score function says which values of theta are best: the optimum score is zero and any non-zero score is less desirable.
	www-cse.ucsd.edu /users/elkan/291winter2004/jan20.html (910 words)

	[Tutor] nested functions
		To rotate point around the x-axis in the XYZ coordinate system, we use the rotation matrix: def xmatrix(theta): return [[1,0, 0], [0, cos(theta),-sin(theta)], [0, sin(theta), cos(theta)]] where theta is the angle you want to rotate by.
		Notice we use a function with theta as an argument.
		Recall that zmatrix is a function, which expects and angle as its argument.
	mail.python.org /pipermail/tutor/2002-April/013688.html (523 words)

	Yorick - Basic Ideas
		As you use a function, you notice that the surrounding code is often the same, suggesting a savings if it were incorporated into the function.
		This does no harm, since any functions or variables defined in the file will be replaced by their new definitions the next time you include the file, just as they would if you typed new definitions.
		The external variable theta determines the phase points at which the damped sine wave is evaluated; q_out will write two header lines, followed by two columns, with one line for each element of the theta array.
	www.cgd.ucar.edu /cms/pjr/yorick/yorickstuff/yorick_1.html (6000 words)

	Exact Formulae for the Lovász Theta Function of Sparse Circulant Graphs
		The Lovász theta function has attracted much attention for its connection with diverse issues such as communicating without errors and computing large cliques in graphs.
		Indeed, this function enjoys the remarkable property of being computable in polynomial time despite being sandwiched between clique and chromatic numbers, two well-known, hard to compute quantities.
		In this paper I provide a closed formula for the Lovász function of all the circulant graphs of degree 4 with even displacement, thus generalizing Lovász results on cycle graphs (circulant graphs of degree 2).
	epubs.siam.org /sam-bin/dbq/article/41852 (160 words)

	Riemann Hypothesis in a Nutshell (Site not responding. Last check: 2007-10-19)
		The functional equation of the zeta function is
		The function Z(t) is typically the object of study for locating zeros of the zeta function on the critical line and verifying the Riemann Hypothesis.
		Since there are infinitely many non-trivial zeros of the zeta function, there is no way you can verify computationally that they all lie on the critical line.
	www.math.ubc.ca /~pugh/RiemannZeta/RiemannZetaLong.html (1384 words)

	Theta function
		With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behaviour with respect to the addition of a period of the associated elliptic functions (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems).
		If τ is fixed, this becomes a Fourier series for a periodic entire function of z, with period one; the theta function satisfying the identity
		If we set z=0 in the above theta functions, we obtain four functions of τ only, defined on the upper half plane (sometimes called theta constants.) These can be used to define a variety of modular forms, and to parameterize certain curves; in particular the Jacobi identity is
	www.askfactmaster.com /Theta_function (382 words)

	Theta functions (Site not responding. Last check: 2007-10-19)
		This counterpart to Poisson's summation formula is shown to be essentially 'equivalent' to the famous functional equation of Riemann's zeta-function, to the 'modular relation' of the theta-function, to the Nielsen-Doetsch summation formula for Bessel functions and to the partial fraction expansion of the periodic Hilbert kernel."
		That the zeta function ought to come into the scheme of things somehow is not surprising - the Poisson and related summation formulae are special cases of the Jacobi theta function.
		Additionally, the properties of the zeta function are automatically connected to the theory of prime numbers.
	www.maths.ex.ac.uk /~mwatkins/zeta/thetafunctions.htm (559 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		function tf = trans_function(theta); % %**************************************************************** % This function performs the transformation from % theta** in [0, pi] to z in [L_END;R_END] using the transformation % % z=.5((R_END-L_END)cos(theta)+L_END+R_END); % % and returns the value f(z), where f is a user specified function.
		% The user must construct an m-file that takes a scaler as % input and gives the scaler value of the function as output.
		% The name of the function must be assigned to the string % variable f.
	amath.colorado.edu /courses/5600/2004fall/pa3/trans_function.m (101 words)

	[No title]
		of the Riemann theta function can be factored * out and the remaining oscillatory part can be approximated with a prescribed error * tolerance.
		* Evaluating the theta function may easily lead to overflows or numerical * instabilities, which should be avoid.
		This is true because * the estimates, which lead to the approximation of the oscillating part of the Riemann * theta function, are rather weak, and therefore the error for the actual values often falls low * of the tolerance.
	www-sfb288.math.tu-berlin.de /~jtem/riemann/src/riemann/theta/Theta.java (1166 words)

	[No title]
		function [THETA,THETA_sum,elapsed_time,keeps_b,Ypred,Ypred_sum] = estim_CEVe(Y,m,H,hor,THETA,flag_beta) % ESTIM_CEVe estimates the parameters of the CEV model as described in % Eraker (2001), using MCMC methods.
		% 'THETA' is a (1 x cT) vector of initial values for parameters, where % cT = 3,4, depending on the model.
		% % OUTPUT VARIABLES % 'THETA' is the (Hr x cT) matrix of draws from parameter posterior distributions.
	galton.uchicago.edu /~artigas/Research/estim_CEVe.m (367 words)

	Solution to Worksheet 8 (Site not responding. Last check: 2007-10-19)
		Both functions are Fortran 90 versions of elemental and pure !
		This is the x component of dB: !___________________________________ function dBx(theta) result(Bx) real(kind=wp), intent(in) :: theta real(kind=wp) :: Bx Bx=zcos(theta)/(1+x2+z2-2xcos(theta))1.5 end function* dBx !___________________________________ !
		This is the x component of B: !___________________________________ function Bx(x_,z_) result(Bx_) real(kind=wp), intent(in) :: x_, z_ real(kind=wp) :: Bx_ real(kind=wp) :: error !
	www.pa.msu.edu /~duxbury/phy201_f00/Worksheet8_Sol.html (257 words)

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