By Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

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**Additional resources for Applied methods of the theory of random functions, Edition: [1st English ed.]**

**Example text**

8) to where X(t) is the function on which the given operator acts, and p(t, tj) is a given function the form of which finally de termines the properties of the operator. In the particular case when the function p(t, fx) is a function of the difference of its arguments, that is, P(*,h) =P(*-h)> formula (8) reduces to the form t Y(t) = jpit-tJXtfJdh. 10) to The importance of operators of this form is due to the fact that the finding of the particular integral of a linear homo geneous equation the right-hand side of which is the func tion X(t) reduces to this.

On the other hand, t0 is arbitrary, con sequently the function p(t, t) must vanish identically. Simi larly it can be shown t h a t the terms outside the integral of all the derivatives of y^t) except the derivative of the n t h order must also vanish. 15) and for the successive derivatives of y^t) we have ' dPj yiW = t C d? -$■ p{t, hWh) &i> i = o, l, . . 16) x(t) + dt71-1 df ti=t j&pit' h)x(h) dt± After combining the integrals into one, the substitution of these relations in the original equations gives f T dn dn~i 1 ^r+%(*) ^ z r + • • • +«*(*) P(*> h)*(h) <*h+ x(t) = x{t).

2 )]}. 31) 56 BANDOM FUNCTIONS where the operators have been provided with the subscripts t± and t2 in order to indicate t h a t in the first case the operator acts with respect to the variable tx and in the second case with respect to the variable t2 and the asterisk above t h e operator shows t h a t if complex expressions occur in the mathematical expression for this operator, it is necessary to replace them by their complex conjugates. Making use of the linearity of the homogeneous operator L and t h e linearity of the operation of finding the mathematical expectation we obtain instead of (31) Ky(tltt2) t h a t is, = M{L?