Tau function information

Tau function may refer to: Tau function (integrable systems), in integrable systems Ramanujan tau function, giving the Fourier coefficients of the Ramanujan...

The Ramanujan tau function, studied by Ramanujan (1916), is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \rightarrow \mathbb {Z} } defined by...

Tau (/ˈtɔː, ˈtɒ, ˈtaʊ/; uppercase Τ, lowercase τ or τ {\displaystyle {\boldsymbol {\tau }}} ; Greek: ταυ [taf]) is the nineteenth letter of the Greek...

f(z)=C\vartheta (z+{\frac {1}{2}}\tau +b,\tau )} for some nonzero C ∈ C {\displaystyle C\in \mathbb {C} } . The Jacobi theta function defined above is sometimes...

( z , 1 , τ ) {\displaystyle \wp (z,\tau ):=\wp (z,1,\tau )} . ℘ {\displaystyle \wp } is a meromorphic function with a pole of order 2 at each period...

Hence we may obtain quantifiers from the choice function, for example P ( τ x ( P ) ) {\displaystyle P(\tau _{x}(P))} was equivalent to ( ∃ x ) ( P ( x )...

{\displaystyle \tau (u)\tau (v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{11}\tau \left({\frac {uv}{\delta ^{2}}}\right),}     where τ(n) is Ramanujan's function.    ...

similar functions, as suggested by tau knockout mice that did not show abnormalities in brain development – possibly because of compensation in tau deficiency...

described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount...

}C_{x}\left(t+{\frac {\tau }{2}},t-{\frac {\tau }{2}}\right)\,e^{-2\pi i\tau f}\,d\tau .} So for a single (mean-zero) time series, the Wigner function is simply given...

More generally, a continuous function ( X , τ X ) → ( Y , τ Y ) {\displaystyle \left(X,\tau _{X}\right)\to \left(Y,\tau _{Y}\right)} stays continuous...

-i\varepsilon }}e^{ix\tau }d\tau .\end{aligned}}} where the second representation is easy to deduce from the first, given that the step function is real and thus...

autocorrelation function of the non-windowed signal x ( t ) {\displaystyle x(t)} , which is denoted as R x x ( τ ) {\displaystyle R_{xx}(\tau )} , provided...

can be expressed as a function of the time-lag, and that this would be an even function of the lag τ = t 2 − t 1 {\displaystyle \tau =t_{2}-t_{1}} . This...

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a function defined as: erf ⁡ z = 2 π ∫ 0 z e − t 2...

{1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar...

arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function). He proved many...

sonar signal processing, an ambiguity function is a two-dimensional function of propagation delay τ {\displaystyle \tau } and Doppler frequency f {\displaystyle...

}^{\infty }\operatorname {rect} (x-\tau )\cdot \operatorname {rect} (\tau )\,d\tau .\\\end{aligned}}} The triangular function can also be represented as the...

In mathematics, the tau conjecture may refer to one of Lehmer's conjecture on the non-vanishing of the Ramanujan tau function The Ramanujan–Petersson...

q_{Y}(\tau )=F_{Y}^{-1}(\tau )=\inf \left\{y:F_{Y}(y)\geq \tau \right\}} where τ ∈ ( 0 , 1 ) . {\displaystyle \tau \in (0,1).} Define the loss function as...

(−1)ω(n), where the additive function ω(n) is the number of distinct primes dividing n. τ(n): the Ramanujan tau function. All Dirichlet characters are...