On the convolution of inverse Gaussian and exponential random variables

We consider the open problem of convolving the first-passage time of a Wiener diffusion process with drift It and variance parameter or 2 with an independent exponential random variable with rate X. We show that if lambda is larger than mu(2)/(2sigma(2)) then the convolution density may be represented using the real and complex parts of the complex error function. Potential applications are briefly mentioned.