TRANSFORMS AND MOMENT GENERATING FUNCTIONS Maurice J. Dupr´ e Department of Mathematics New Orleans, LA 70118 email: mdupre@tulane.edu November 2010 There are several operations that can be done on functions to produce new functions in ways that can aid in various calculations and that are referred to as transforms. The general setup is a transform operator T which is really a function whose domain is in fact a set of functions and whose range is also a set of functions. You are used to thinking of a function as something that gives a numerical output from a numerical input. Here, the domain of T is actually a set D of functions and for f ∈D the output of T is denoted T (f ) and is in fact a completely new function. A simple example would be differentiation, as the differentiation operator D when applied to a differentiable function gives a new function which we call the derivative of the function. Likewise, antidifferentiation gives a new function when applied to a function. As another example, we have the Laplace transform, L, which is defined by [L(f )](s)=  ∞ 0 e −sx f (x)dx. With a general transform, it is best to use a different symbol for the independent variable of the transformed function than that used for the independent variable of the original function in applications, so as to avoid confusion. The idea of the Laplace transform is that it has useful properties for solving differential equations-it turns them into polynomial equations which can then be solved by algebraic methods. A related transform is the Fourier transform, F , which is defined by [F (f )](t)=  ∞ −∞ e itx f (x)dx. More generally, we can take any function h of two variables, say w = h(s, x) and for any function f viewed as a function of x, we can integrate h(s, x)f (x) with respect to x and the result is a function of s alone. Thus we can define the transform H by the rule [H(f )](s)=  ∞ −∞ h(s, x)f (x)dx. A transform like H is called an Integral Transform as it primarily works through integration. In this situation, we refer to h as the Kernel Function of the transform. Thus, for the Laplace transform L and the Fourier transform F , the kernel function is an exponential function. Keep in mind that integration usually makes functions smoother. For instance antidifferentiation certainly increases differentiability. In general, to make good use of a transform you have to know its properties. A very useful property which the preceding examples are easily seen to have is Linearity: T (a · f ± b · g)= a ·T (f ) ± b ·T (g), a,b ∈ R, f,g ∈D. A transform which has the property of being linear is called a linear transform, and we see that for a linear transform, to find the transform of a function which is itself a sum of functions we just work out the transform of each term and add up the results. Notice that the Fourier and 1