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# Thermal Gradients in the Complex Plane

This research will investigate the relationship between heat gradients and complex analysis. We have established a connection between the steady-state heat gradient equation and Cauchy’s Integral Theorem that will allow us to apply complex analysis to a heat gradient in order to estimate the temperature inside the gradient. We believe that this offers an easier alternative method to mapping the thermal gradient inside an object, even in complicated heat source arrangements.

This will be experimentally done by drawing a grid on a metal sheet and applying a hot reservoir to one side and a cold reservoir on the other, creating our heat gradient. Temperatures will be measured along the outside of the grid and at an interior point until steady state is achieved. At this point, Cauchy’s Integral Theorem will be used to calculate the theoretical temperature at the interior point. This calculated temperature will then be compared to the actual temperature to find our error. The number of measurements along the outside of the grid will be adjusted until our error is reached.

Our goal for this project is to determine the minimum number of temperature measurements we can take and apply the Cauchy Integral to, in order to calculate the temperature of any point on the gradient within a certain error.

This paper will focus in on the experiment and the analysis of the results received from it. Included in this will be the theoretical explanation for using this method, details of the experiment, data from the experiment, and associated code and images from the models produced.