Sentence examples for integral follows from inspiring English sources

The calculation of the first integral follows from an application of the theory of order statistics (cf. [12, eq. (17)]) to obtain an expression for (f_{X_{k}}).

The above definition corresponds to the one for Riemann Liouville fractional integral of order (q>0) when (rho=1), while the famous Hadamard fractional integral follows for (rhorightarrow0), that is, lim_{rhorightarrow0},^{rho}I^{q}f(t)= frac{1}{Gamma q }int_{0}^{t} biggl( logfrac{t}{s} biggr) ^{q-1}frac{f(s)}{s},ds. ([4]).

Remark 8 Notice that from (2.11) in fact the value of AW integral follows since we see that ∫ − 1 1 f A W ( x | a, b, c, d | q ) d x = 1, which means that the integral 1 2 π ∫ − 1 1 1 1 − x 2 ∏ n ≥ 0 l ( x | q n ) v ( x | a q n ) v ( x | b q n ) v ( x | c q n ) v ( x | d q n ) d x = ( a b c d ) ∞ ( q, a b, a c, a d, b c, b d, c d ) ∞. (2.14).

The first one is obtained from a line integral followed by Cauchy's residue theorem, and expressed in terms of Stroh's eigenvalues.

The double inequality of equation (31) containing the integrals follows from the inequality in equation (12).

Based on the Liu process, a Liu integral is defined as a fuzzy counterpart of an Ito integral as follows: [32].

The identity (7) can also be written in the form of an integral as follows.

For, we define -Euler polynomials of Apostol's type using the fermionic -adic integral as follows: (2.12).

Since there is no additional Jacobian factor in the Onsager Machlup path integral, it follows that the Langevin equation is of the Ito form.

A generalized closed-form expression for the moments of the SNR can be obtained by substituting (25) into and evaluating the resulting integral as follows (Appendix G): (28).

For a pair ((w(t), X t))) of a Wiener process (w(t)) and random process (X t)), we define the Itô integral as follows: I(X) = int_{0}^{T}X t),dw(t).