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...

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...

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...

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...

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...

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...

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...

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...

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

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...

{\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...

}{\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...

{\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...

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...

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...

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...

}(\sigma )} , U τ ( τ ) {\displaystyle U_{\tau }(\tau )} , and U z ( z ) {\displaystyle U_{z}(z)} are arbitrary functions. Substitution of the completely separated...

between Seiberg–Witten theory and integrable systems has been reviewed by Eric D'Hoker and D. H. Phong. See Hitchin system. Using supersymmetric localisation...

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

(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)...

{\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...

_{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...

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

1+\tau {}\alpha } ) to the term itself. Via substitution and arithmetic, the type expands to 1 + τ + τ 2 + τ 3 + ⋯ {\displaystyle 1+\tau +\tau ^{2}+\tau...

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...

}u_{_{P}}(x-\tau )\cdot v(\tau )\ d\tau \\&\equiv \int _{P}u_{_{P}}(x-\tau )\cdot v_{_{P}}(\tau )\ d\tau ;\quad \quad \scriptstyle {\text{integration over any interval...

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...

{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...