ON A FAMILY OF LOGARITHMIC AND EXPONENTIAL INTEGRALS OCCURRING IN PROBABILITY AND RELIABILITY THEORY

We define an integral function I(mu)(alpha, x; a, b) for non-negative integral values of mu by I(mu)(alpha, x; a, b) = integral-infinity/x (In t)mu t(alpha-1) exp (-at - bt-1) dt (x greater-than-or-equal-to 0, a > 0, b > 0, -infinity < alpha < infinity). It is proved that I(mu)(alpha, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.