Find an image of a given function. How to find a similar picture, photograph, image on the Internet

The problem is posed as follows: given a function F(p), we need to find the function /(<)>whose image is F(p). Let us formulate conditions sufficient for the function F(p) of a complex variable p to serve as an image. Theorem 12. If a function F(p) analytic in a half-plane 1) tends to zero for any half-plane Rep = a > s0 uniformly with respect to arg Finding the original from an image 2) the integral a-xu converges absolutely, then F(p) is an image some original function f(t). Tasks*. Can the function F(p) = ^ serve as an image of some original function? We will indicate some ways to find the original from an image. 3.1. Finding the original using image tables First of all, it is worth bringing the function F(p) to a simpler, “tabular” form. For example, in the case when F(p) is a fractional rational function of the argument p, it is decomposed into elementary fractions and the appropriate properties of the Laplace transform are used. Example 1. Find the original for We write the function F(p) in the form Using the displacement theorem and the linearity property of the Laplace transform, we obtain Example 2. Find the original for the function M We write F(p) in the form Hence / 3.2. Using the inversion theorem and its corollaries Theorem 13 (inversion). /Gauche function fit) is the original function with growth exponent s0 and F(p) is its image, then at any point of continuity of the function f(t) the relation is satisfied where the integral is taken along any straight line and is understood in the sense of the principal value, i.e. . as Formula (1) is called the Laplace transform inversion formula, or Mellin's formula. In fact, let, for example, f(t) be piecewise smooth on each finite segment)