In This Paper Three Series Expansions Are Derived For The Wave Function of A Vertical Dipole Placed At The Surface of A Plane Earth. Two Convergent Series And One Asymptotic Series Are Obtained. A Remainder Term For The Latter Series Is Given Which Enables One To Set An Upper Limit To The Amount of Error Obtained By Stopping At Any Particular Stage In The Series.
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S. O. Rice (1937) studied this question.