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Special functions and their transforms. Gamma function. Bessel functions. Error function. Complementary Error function. Exponential integral. Sine and cosine integrals. Dirac delta function. Unit Step Function. Null function.

Def. Gamma function. For positive values of x, the Gamma function is defined by the improper integral

This is Euler’s second integral. It converges for all positive, real values of x. An important property of the gamma function is the relationship

1)        Γ(x + 1) = xΓ(x) .

The gamma function can be defined for negative values of x using 1) i.e.

Bessel functions of the first kind. Let ν be a real number. The function

is known as the Bessel function of the first kind of order ν. The formula is valid providing ν -1, -2, -3, .... . Γ(ν) is the gamma function.

Error function. The error function is defined as

Complementary Error function. The complementary error function is defined as

Exponential integral. The exponential integral is defined as

Sine integral. The sine integral is defined as

Cosine integral. The cosine integral is defined as

Dirac delta function. The Dirac delta function δ(t), also called the impulse function, is a function which is zero for every value of t except at t = 0, where it is infinite in such a way that

Consider the function Fε(t)

consisting of a pulse of width ε and height 1/ε shown in Fig. 1. The Dirac delta function can be viewed as the limit of Fε(t) as the width ε of the pulse approaches zero. Two of its properties are

The Dirac delta function is not, strictly speaking, a function but it is treated as a function, manipulated as a function.

Unit Step Function. The unit step function is defined as

Syn. Heaviside’s unit function

Fig. 1a shows a graph of y = u(t). Fig. 1b shows a graph of y = u(t - a)

Remark. The function g(t) = f(t)u(t - a) represents the function f(t) as cut off to the left of t = a.. See Fig. 3.

Null function. If N(t) is a function of t such that for all t > 0

we call N(t) a null function.

Example. The function

is a null function.

In general, a function that is zero at all but a countable set of points is a null function.

Transforms of special functions

 F(t) f(s) = L[F(t)] 1 (ν real) 2 3 4 5 6 erf(t) 7 8 Si(t) 9 Ci(t) 10 Ei(t) 11 u(t - a) 12 1 13 14 N(t) 0

References

Murray R. Spiegel. Laplace Transforms. (Schaum)

Murray R. Spiegel. Mathematical Handbook. (Schaum)

C. R. Wylie, Jr. Advanced Engineering Mathematics.