Geometric Mapping of the Complex Exponential Function exp(z)

This applet visually demonstrates the geometric mapping properties of the complex exponential function w = exp(z). Users can interact with the applet by dragging points on the complex z-plane to observe how individual complex numbers are transformed under the exponential function into the w-plane. The central feature of the applet is an orthogonal grid superimposed on the complex z-plane, consisting of horizontal lines (where the imaginary part of z is constant) and vertical lines (where the real part of z is constant). As these grid lines are mapped through the function w = exp(z), learners can clearly see that horizontal lines transform into rays emanating from the origin in the w-plane, while vertical lines transform into concentric circles centered at the origin. This transformation vividly illustrates the formula exp(x + iy) = e^x · (cos y + i sin y), showing how the real part x controls the modulus (distance from the origin) and the imaginary part y controls the argument (angle). The applet highlights the conformal nature of the complex exponential function: because the original grid lines intersect at right angles and their images also intersect at right angles, the mapping preserves angles locally, which is the defining property of a conformal map. Students can explore multiple visual aids simultaneously, including animated traces, color-coded correspondences between original and image curves, and toggles for showing or hiding specific grid families. By manipulating the visible domain or animation speed, learners gain deeper intuition about periodicity, the role of the imaginary period 2πi, and how the exponential function wraps the infinite horizontal strip of the z-plane onto the entire punctured w-plane. This interactive exploration strengthens conceptual understanding of complex analysis topics such as conformal mapping, the geometric meaning of the complex exponential, and the relationship between algebraic formulas and their visual representations, making it a valuable teaching tool for undergraduate mathematics courses.