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Mathematical function
mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties
Ramanujan_theta_function
Indian mathematician (1887–1920)
unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas
Srinivasa_Ramanujan
Special functions of several complex variables
mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode
Theta_function
Mathematical identities related to integer partitions
the following identities to the remaining Rogers–Ramanujan functions and to the Ramanujan theta function described above: S ( q ) = q 1 / 5 H ( − q ) G
Rogers–Ramanujan_identities
q-Pochhammer symbol. elliptic hypergeometric series Jacobi theta function Ramanujan theta function Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric
Q-theta_function
Mathematical function
functions. Elliptic curve Schwarz–Christoffel mapping Carlson symmetric form Jacobi theta function Ramanujan theta function Dixon elliptic functions Abel
Jacobi_elliptic_functions
Figurate number
the sum of triangular numbers are connected to theta functions, in particular the Ramanujan theta function. The number of line segments between closest
Triangular_number
Complex-differentiable part of a Maass wave function
theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in
Mock_modular_form
Topics referred to by the same term
1/2 Ramanujan theta function, f ( a , b ) {\displaystyle f(a,b)} Neville theta functions Riemann–Siegel theta function, θ ( t ) {\displaystyle \theta (t)}
Theta function (disambiguation)
Theta_function_(disambiguation)
Inverse functions of sin, cos, tan, etc.
trigonometric functions. For example, if x = sin θ {\displaystyle x=\sin \theta } , then d x / d θ = cos θ = 1 − x 2 , {\textstyle dx/d\theta =\cos \theta ={\sqrt
Inverse trigonometric functions
Inverse_trigonometric_functions
connections and consistent notation with the Ramanujan G- and g-functions and the Jacobi theta functions, both of which conventionally uses the nome.
Weber_modular_function
Collection of Srinivasa Ramanujan's discoveries in mathematics
was settled because Ramanujan's final letters to Hardy had referred to the discovery of what Ramanujan called mock theta functions, although without great
Ramanujan's_lost_notebook
Mathematical constants
related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. ISSN 1572-9303. Weisstein, Eric W. "Gamma Function". MathWorld. Raimundas
Particular values of the gamma function
Particular_values_of_the_gamma_function
identity Ramanujan machine Ramanujan–Nagell equation Ramanujan–Peterssen conjecture Ramanujan–Soldner constant Ramanujan summation Ramanujan theta function Ramanujan
List of things named after Srinivasa Ramanujan
List_of_things_named_after_Srinivasa_Ramanujan
define the function E ( x ) = ∫ 0 π / 2 1 − x sin 2 θ d θ , {\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x\sin ^{2}\theta }}\ d\theta ,} known as
Perimeter_of_an_ellipse
Number of partitions of an integer
this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial
Partition function (number theory)
Partition_function_(number_theory)
Mathematical techniques for summing divergent infinite series
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan
Ramanujan_summation
Special function defined by an integral
elliptic functions Jacobi theta function Meridian arc Pendulum period Ramanujan theta function Schwarz–Christoffel mapping Weierstrass's elliptic functions K
Elliptic_integral
Technique in analytic number theory
work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function. It was taken up by many other
Hardy–Ramanujan–Littlewood circle method
Hardy–Ramanujan–Littlewood_circle_method
Function whose domain is the positive integers
Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. An arithmetic function a is
Arithmetic_function
Elliptic analog of hypergeometric series
modified Jacobi theta function with argument x and nome p is defined by θ ( x ; p ) = ( x , p / x ; p ) ∞ {\displaystyle \displaystyle \theta (x;p)=(x,p/x;p)_{\infty
Elliptic hypergeometric series
Elliptic_hypergeometric_series
Mathematical function
Jacobi Theta function and ϑ 1 ( z | τ ) = − ϑ 11 ( z ; τ ) {\displaystyle \vartheta _{1}(z|\tau )=-\vartheta _{11}(z;\tau )} Because the eta function is easy
Dedekind_eta_function
Mathematical functions
exponential function. An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see Lemniscate
Lemniscate_elliptic_functions
Continued fraction closely related to the Rogers–Ramanujan identities
related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument. Given the functions G ( q ) {\displaystyle
Rogers–Ramanujan continued fraction
Rogers–Ramanujan_continued_fraction
Nearest integers from a number
" Ramanujan, Question 723, Papers p. 332 Somu, Sai Teja; Kukla, Andrzej (2022). "On some generalizations to floor function identities of Ramanujan" (PDF)
Floor_and_ceiling_functions
Probability distribution
approximation of the median by comparing the median to Ramanujan's θ {\displaystyle \theta } function. Berg and Pedersen found more terms: ν ( α ) = α − 1
Gamma_distribution
Mathematical identity found by Jacobi in 1829
enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For | a b | < 1 {\displaystyle |ab|<1} it can be written as ∑
Jacobi_triple_product
Approximation for factorials
} An alternative approximation for the gamma function stated by Srinivasa Ramanujan in Ramanujan's lost notebook is Γ ( 1 + x ) ≈ π ( x e ) x ( 8 x
Stirling's_approximation
Analytic function in mathematics
Particular values of the Riemann zeta function Prime zeta function Renormalization Riemann–Siegel theta function ZetaGrid "Jupyter Notebook Viewer". Nbviewer
Riemann_zeta_function
Mathematical concept
result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be
Infinite_product
Landau–Ramanujan constant, Mock theta functions, Ramanujan conjecture, Ramanujan prime, Ramanujan–Soldner constant, Ramanujan theta function, Ramanujan's sum
Timeline_of_Indian_innovation
S2CID 119142457 Ismail, M. E. H.; Zhang, R. (2018b), "q-Bessel Functions and Rogers-Ramanujan Type Identities", Proceedings of the American Mathematical Society
Jackson_q-Bessel_function
Dutch mathematician (born 1975)
for making a connection between Maass forms and Srinivasa Ramanujan's mock theta functions in 2002. He was born in Oosterhout. After a period at the Max-Planck
Sander_P._Zwegers
Number, approximately 3.14
. An example is the Jacobi theta function θ ( z , τ ) = ∑ n = − ∞ ∞ e 2 π i n z + π i n 2 τ , {\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty
Pi
List of notable people who belong to the Brahmin caste
Srinivasa Ramanujan, Greatest Indian mathematician who compiled Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions and
List_of_Brahmins
Series representing modular forms
on modular invariants provides expressions for these two functions in terms of theta functions. Any holomorphic modular form for the modular group can
Eisenstein_series
Class of mathematical functions
\eta } is the Dedekind eta function. For the Fourier coefficients of Δ {\displaystyle \Delta } , see Ramanujan tau function. e 1 {\displaystyle e_{1}}
Weierstrass_elliptic_function
Uses of the constant
Mathematical Society. ISBN 0-8218-3246-8. p. 112 Cooper, Shaun (2017). Ramanujan's Theta Functions (First ed.). Springer. ISBN 978-3-319-56171-4. (page 647) Euler
List_of_formulae_involving_π
Indian inventions
Kesavan Raghavan Nair in 1939. Ramanujan theta function, Ramanujan prime, Ramanujan summation, Ramanujan graph and Ramanujan's sum – Discovered by the Indian
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
Expression which is not assigned an interpretation
However, Ramanujan summation is useful for modelling a number of real-world phenomena, including the Casimir effect and bosonic string theory. A function may
Undefined_(mathematics)
Analytic function on the upper half-plane with a certain behavior under the modular group
essentially Ramanujan's mock theta functions. Groups which are not subgroups of SL(2, Z) can be considered. Hilbert modular forms are functions in n variables
Modular_form
Special mathematical function
description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents
Nome_(mathematics)
Symmetric holomorphic function
)=k^{2}(\tau )} . In terms of the Dedekind eta function η ( τ ) {\displaystyle \eta (\tau )} and theta functions, λ ( τ ) = ( 2 η ( τ 2 ) η 2 ( 2 τ ) η 3 (
Modular_lambda_function
Mathematical term
}q^{n^{2}}} with the sum on the right similar to the Ramanujan theta function, or Jacobi theta function ϑ 3 ( q ) {\displaystyle \vartheta _{3}(q)} . Note
Lambert_series
Conjecture on zeros of the zeta function
finite graph is a Ramanujan graph, a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue
Riemann_hypothesis
Mathematical constant
/2}e^{\pi \tan \theta }e^{-e^{\pi \tan \theta }}\,d\theta .} The Fransén–Robinson constant can also be expressed using the Mittag-Leffler function as the limit
Fransén–Robinson_constant
Srinivasa Ramanujan, Indian mathematician – Ramanujan prime, Ramanujan theta function, Ramanujan's sum, Ramanujan's master theorem, Landau–Ramanujan constant
List_of_eponyms_(L–Z)
Modular function in mathematics
found in Ramanujan's theory of elliptic functions to alternative bases. The inversion is applied in high-precision calculations of elliptic function periods
J-invariant
Jackson integral Carl Gustav Jakob Jacobi: Jacobi polynomial, Jacobi theta function Joseph Marie Kampe de Feriet (1893–1982): Kampe de Feriet hypergeometric
List of eponyms of special functions
List_of_eponyms_of_special_functions
Topics referred to by the same term
moonshine, a mysterious connection between Niemeier lattices and Ramanujan's mock theta functions Francisco Umbral (1932–2007), Spanish journalist, novelist
Umbral
Discrete probability distribution
v)=\exp[(\theta _{1}-\theta _{12})(u-1)+(\theta _{2}-\theta _{12})(v-1)+\theta _{12}(uv-1)]} with θ 1 , θ 2 > θ 12 > 0 {\displaystyle \theta _{1},\theta _{2}>\theta
Poisson_distribution
Special function in mathematics
representation along with the residue theorem. A second proof uses a theta function identity, or equivalently Poisson summation. These proofs are analogous
Hurwitz_zeta_function
Mathematical conjecture about zeros of L-functions
for much more general L-functions than Dedekind zeta functions lie on critical lines. One example can be Ramanujan L-function related to modular form
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Constant e raised to the power of pi
Doman in September 2023 and is a result of a sum related to Jacobi theta functions as follows: ∑ k = 1 ∞ ( 8 π k 2 − 2 ) e − π k 2 = 1. {\displaystyle
Gelfond's_constant
Chinese professor of mathematics
contributions span a wide spectrum of topics such as arithmetic theta lifts and derivatives of L-functions, the Gan–Gross–Prasad conjecture and its arithmetic counterpart
Yifeng_Liu
Special mathematical function
(1989). "The dilogarithm function in geometry and number theory". Number Theory and Related Topics: papers presented at the Ramanujan Colloquium, Bombay, 1988
Polylogarithm
Concept in combinatorics (part of mathematics)
Pentagonal number theorem q-derivative q-theta function q-Vandermonde identity Rogers–Ramanujan identities Rogers–Ramanujan continued fraction Berndt, B. C. "What
Q-Pochhammer_symbol
Mathematical operation
{\displaystyle F(s,\theta )=A(s){\frac {\sin(s(\theta _{0}-\theta ))}{\sin(2\theta _{0}s)}}+B(s){\frac {\sin(s(\theta _{0}+\theta ))}{\sin(2\theta _{0}s)}}} Now
Mellin_transform
Difference between logarithm and harmonic series
generalized-Euler-constant function and its derivative, arXiv:0808.0410 Berndt, Bruce C. (January 2008). "A fragment on Euler's constant in Ramanujan's lost notebook"
Euler's_constant
Axiomatic definition of a class of L-functions
is entire function in S, then F ( s + i t ) {\textstyle F(s+it)} for t ∈ R {\textstyle t\in \mathbb {R} } is also in S. From the Ramanujan conjecture
Selberg_class
Nineteenth letter in the Greek alphabet
chronic traumatic encephalopathy Divisor function in number theory, also denoted d or σ0 Ramanujan tau function Golden ratio (1.618...), although φ (phi)
Tau
English mathematician
'Srinivasa Ramanujan" Nature 149:292. 1944: Jacobian Elliptic Functions, Clarendon Press via Internet Archive Neville's algorithm Neville theta functions Senechal
Eric_Harold_Neville
German mathematician (born 1977)
Germany, who has made fundamental contributions to the theory of mock theta functions. Kathrin Bringmann was born on 8 May 1977, in Muenster, Germany. She
Kathrin_Bringmann
who exhibited the zeta function as the Mellin transform of Jacobi's theta function. Riemann used asymptotics of the theta function to obtain the analytic
Rankin–Selberg_method
Type of generalization of periodic functions in Euclidean space
symplectic group, arose naturally from considering moduli spaces and theta functions. The post-war interest in several complex variables made it natural
Automorphic_form
Relation between sides of a right triangle
\theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}\right)\\&=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos \left(\theta _{1}-\theta
Pythagorean_theorem
English mathematician (1886–1965)
years on Ramanujan's formulae in the area of modular equations, mock theta functions and q-series, and for some time looked after Ramanujan's lost notebook
G._N._Watson
Topic in group theory and harmonic analysis (Niemeier lattice-mock theta connection)
moonshine is a mysterious connection between Niemeier lattices and Ramanujan's mock theta functions. It is a generalization of the Mathieu moonshine phenomenon
Umbral_moonshine
"generalized Ramanujan conjecture" for (quasi-) split groups", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc
Θ10
Concept in geometry
{r^{2}\left(1-\sin ^{2}\theta \right)}}\cdot r\cos \theta \,d\theta \\[5pt]&=2r^{2}\int _{0}^{\frac {\pi }{2}}\cos ^{2}\theta \,d\theta \\[5pt]&={\frac {\pi
Area_of_a_circle
Indian mathematician and professor (born 1972)
his Ph.D. thesis was Contributions to Ramanujan's Schlafli-type Modular Equations, Class Invariants, Theta-functions, and Continued Fractions. Following
Nayandeep_Deka_Baruah
& P. Flajolet (2010) “Pseudo-factorials, elliptic functions, and continued fractions” The Ramanujan journal 21(1), 71–97. https://arxiv.org/pdf/0901.1379
Dixon_elliptic_functions
Plane curve
\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta } which is in general not an elementary function. The circumference of the ellipse may be evaluated
Ellipse
Equation in Fourier analysis
to prove the functional equation for the theta function. Poisson's summation formula appears in Ramanujan's notebooks and can be used to prove some of
Poisson_summation_formula
Infinite product identity introduced by Watson
W. N. (1951), "On the simplification of some identities of the Rogers-Ramanujan type", Proceedings of the London Mathematical Society, Third Series, 1:
Quintuple_product_identity
Symbols for constants, special functions
correlation coefficient, a measure of rank correlation in statistics Ramanujan's tau function in number theory shear stress in continuum mechanics a type variable
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Sum of inverse squares of natural numbers
archived from the original (PDF) on 2011-07-06 Berndt, Bruce C. (1989), Ramanujan's Notebooks: Part II, Springer-Verlag, p. 150, ISBN 978-0-387-96794-3 An
Basel_problem
Summation formula in Mathematics
(x+1)=\operatorname {Li} _{-x}\left(e^{-1}\right)+\theta (x)} where Γ ( x ) {\displaystyle \Gamma (x)} is the gamma function, Li s ( z ) {\displaystyle \operatorname
Abel–Plana_formula
24-dimensional repeating pattern of points
\sigma _{11}(n)} is the divisor function for exponent 11, and τ ( n ) {\displaystyle \tau (n)} is the Ramanujan tau function. It follows that for m ≥ 1, the
Leech_lattice
MathWorld. Weisstein, Eric W. "Landau-Ramanujan Constant". MathWorld. Weisstein, Eric W. "Nielsen-Ramanujan Constants". MathWorld. Weisstein, Eric W
List of mathematical constants
List_of_mathematical_constants
Indian mathematician and astronomer (1340–1425)
⋯ {\displaystyle r\theta ={\frac {r\sin \theta }{\cos \theta }}-(1/3)\,r\,{\frac {\left(\sin \theta \right)^{3}}{\left(\cos \theta \right)^{3}}}+(1/5)\
Madhava_of_Sangamagrama
Mathematical function
Riesz function is defined on the strip − 1 < ℜ ( s ) < − 1 2 {\displaystyle -1<\Re (s)<-{\frac {1}{2}}} . On this strip, we have (cf. Ramanujan's master
Riesz_function
Solved prime-number problem
proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan. The following elementary proof was published by Paul Erdős in 1932, as
Proof_of_Bertrand's_postulate
Sequence of polynomials
}e^{x{\bigl (}e^{\cos(\theta )}\cos(\sin(\theta ))-1{\bigr )}}\cos {\bigl (}xe^{\cos(\theta )}\sin(\sin(\theta ))-n\theta {\bigr )}\,d\theta \,.} Bell polynomials
Touchard_polynomials
Fictional book from the Sherlock Holmes book series
genius Ramanujan. Scribner. p. 168. ISBN 978-0-671-75061-9. Watson, G. N. (2001). "The final problem: an account of the mock theta functions". Ramanujan: essays
The_Dynamics_of_an_Asteroid
JSTOR 1990319. MR 0011087. Bertrand, Daniel (1997). "Theta functions and transcendence". The Ramanujan Journal. 1 (4): 339–350. doi:10.1023/A:1009749608672
Four_exponentials_conjecture
3160290618 Altug, Salim Ali; Tsimerman, Jacob (2010). "Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms". International
Waldspurger_formula
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
{\displaystyle k_{\theta }={\begin{pmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\\\end{pmatrix}}\in SO(2),\theta \in \mathbb {R} .}
Maass_wave_form
American mathematician
from it) became one of only three works to study the mock theta functions between Ramanujan in the 1920s and the work of George Andrews beginning in 1966
Leila_Bram
Ramanujan develops over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions
Timeline_of_number_theory
Algorithmic runtime requirements for common math procedures
(1988). "Approximations and complex multiplication according to Ramanujan". Ramanujan revisited: Proceedings of the Centenary Conference. Academic Press
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Perimeter of a circle or ellipse
S2CID 126427943. Almkvist, Gert; Berndt, Bruce (1988), "Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary", American
Circumference
In mathematics, a non-algebraic number
73.140. ISSN 0386-2194. Bertrand, Daniel (1997). "Theta functions and transcendence". The Ramanujan Journal. 1 (4): 339–350. doi:10.1023/A:1009749608672
Transcendental_number
Constants in the zeta function's Laurent series expansion
infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some
Stieltjes_constants
Russian mathematician (1937–2008)
Applying his p {\displaystyle p} -adic form of the Hardy-Littlewood-Ramanujan-Vinogradov method to estimating trigonometric sums, in which the summation
Anatoly_Karatsuba
16th-century Sanskrit treatise on astronomy
well known result sin θ ≈ θ {\displaystyle \sin \theta \approx \theta } when θ {\displaystyle \theta } is in radians and is small. Full text of the work
Grahalaghava
Number, approximately 1.46557
= xn/m German Wikipedia has a table of analytical values of the Ramanujan G-function [de] for odd arguments below 47. Sloane, N. J. A. (ed.). "Sequence
Supergolden_ratio
Mathematical technique
Arthur Cayley, Schröder, Ramanujan, Riordan, Knuth, Comtet [fr], etc.). It was then slowly realized that the generating functions were capturing many other
Symbolic method (combinatorics)
Symbolic_method_(combinatorics)
British mathematician (1866–1956)
the theory of the theta functions (Cambridge: The University Press, 1897) An introduction to the theory of multiply periodic functions (Cambridge: The University
Henry_F._Baker
Theory of a class of elliptic curves
38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974. Ramanujan Constant – from Wolfram MathWorld Silverman 1986, p. 339. Silverman 1994
Complex_multiplication
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