Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other...

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

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...

characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution)...

g exists if f and g are both Lebesgue integrable functions in L1(Rd), and in this case f∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This...

Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals...

they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite...

curves and surfaces. Every continuous function f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } is integrable (for example in the sense of the Riemann...

energy is finite (i.e. x ( t ) {\displaystyle x(t)} is a square-integrable function) allows applying Parseval's theorem (or Plancherel's theorem). That...

}{\frac {u(t-\tau )-u(t+\tau )}{2\tau }}\,\mathrm {d} \tau .} When the Hilbert transform is applied twice in succession to a function u, the result is...

stochastic random functions are usually not absolutely integrable. Nor is r x x {\displaystyle r_{xx}} assumed to be absolutely integrable, so it need not...

{\displaystyle \tau } . If f {\displaystyle f} and g {\displaystyle g} are both continuous periodic functions of period T {\displaystyle T} , the integration from...

probability density function is given by[citation needed] p ( τ ∣ λ , β )   d τ = λ Γ ( 1 + β − 1 )   e − ( τ λ ) β   d τ {\displaystyle p(\tau \mid \lambda...

{\left({\frac {1}{2}}\right)}}}\int _{0}^{\infty }\tau (\tau -1)\cdots (\tau -n+1)\tau ^{-{\frac {1}{2}}}e^{-\tau }\,d\tau \\[1ex]&=\sum _{k=0}^{n}\left({\frac {1}{2}}\right)^{\bar...

The number 𝜏 (/ˈtaʊ, ˈtɔː, ˈtɒ/ ; spelled out as tau) is a mathematical constant that is the ratio of a circle's circumference to its radius. It is approximately...

of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞). For locally integrable functions...

{\displaystyle \tau } is the variable of integration (takes on values from time 0 to the present t {\displaystyle t} ). Equivalently, the transfer function in the...

autocorrelation function: p.395  R X X ⁡ ( τ ) = E ⁡ [ X t + τ X ¯ t ] {\displaystyle \operatorname {R} _{XX}(\tau )=\operatorname {E} \left[X_{t+\tau }{\overline...

(n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{0}^{t}(t-\tau )^{n-q-1}f(\tau )\,d\tau +\Psi (x)} Initial conditions Dynamical systems Lorenzo, Carl F.; Hartley, Tom T. (2000)...

transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier...

{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...

) {\displaystyle L^{1}([0,P])} function which is periodic on R {\displaystyle \mathbb {R} } , and therefore integrable on any interval of length P . {\displaystyle...

{L}}(\gamma (\tau ;\cdot ),{\dot {\gamma }}(\tau ;\cdot ),\tau )\,d\tau ,} where γ = γ ( τ ; t , t 0 , q , q 0 ) , {\displaystyle \gamma =\gamma (\tau ;t,t_{0}...

studied in the two-dimensional case as an integrable system, in particular an integrable field theory. Its integrability was shown by Faddeev and Reshetikhin...

_{t_{0}}^{t}I(\tau )\,\mathrm {d} \tau } More sophisticated current integrator circuits build on this relation, such as the charge amplifier. A current integrator is...

{\displaystyle \tau } and τ m {\displaystyle \tau _{m}} as well as the coupling parameters a and b. The adaptive exponential integrate-and-fire model inherits...

τ ) {\displaystyle \psi (\mathbf {x} ,\tau )} .] In real time, the 2 n {\displaystyle 2n} -point Green function is defined by G ( n ) ( 1 … n ∣ 1 ′ … n...

mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet...

do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in...

symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which...