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In this paper, we calculate the transient eddy-current response of a driver pickup probe due to a conducting plate by means of series expansions. We consider an excitation current in the driver coil that rises exponentially with time toward a constant value, the step function current being a special case with zero time constant. We find the transient response from the frequency domain expressions for the induced electromotive force in the pickup coil by using the inverse Laplace transform. Usually, the field of the excitation coil is expressed in the form of a Bessel integral with respect to the radial spatial frequency. Here, however, to improve computational efficiency, we approximate the integral as a series expansion by truncating the solution domain at a large but finite radius and setting the field on the truncation boundary to zero. The series is a sum of terms containing discrete spatial frequencies defined by the zeros of the first-order Bessel function of the first kind. We then form an inner summation from the temporal frequency-dependent terms in order to facilitate an analytical inverse Laplace transformation to the time domain. We present two alternative expansions for the inner sum. One is constructed with reference to the short time regime and the other with respect to the long time limit. Numerical results show good agreement between the alternative series approximations at a wide range of intermediate times.