Space weather storms involve intense and rapidly varying electric currents in the ionosphere, which create electric and magnetic fields at the Earth's surface. The electric fields drive geomagnetically induced currents (GIC) in technological networks and may have serious impacts. For assessing the hazards it is necessary to estimate GIC magnitudes, and this requires calculations of the electric and magnetic fields produced at the Earth's surface by the ionospheric currents. The surface fields are also affected by currents induced within the ground and influenced by the conductivity of the Earth. This also has to be taken into account. The calculation methods should be fast enough that they can be applied to forecasting the fields and GIC, for example, by using satellite observations of the solar wind. In this paper, we consider an infinitely long horizontal line current, which is the basic model of an auroral electrojet and simple enough to give insight into the physics and calculation techniques. The Earth is assumed to be composed of horizontal layers. We consider the exact integral expressions of the fields at the Earth's surface. The applicability of a series expansion technique (SER) and the complex image method (CIM), both of which were originally developed for other disciplines, are reviewed and summarized by giving the expressions of the electric and magnetic fields at the Earth's surface and by considering the mathematical assumptions required. Numerical calculations and comparisons with exact solutions show that SER and CIM are very accurate.