| | Category | MA | P24 | Converging Logarithmic Spirals on the Complex Plane |
| | Abstract | The iteration of the logarithmic function on a complex number is rather |
| | complicated to calculate by hand; however, using computer programs to |
| | mathematically simplify the logarithmic operations gives us data that |
| | suggests that iterating the logarithmic function on any complex number will |
| | eventually reach a unique limit no matter what the initial inputted value is. |
| | My project revolves around giving a rigorous mathematical proof to explain |
| | why this convergence occurs. Manipulating common equations in Complex |
| | Analysis and applying the Mean Value Theorem on the complex plane |
| | shows not only that the limit does exist but also shows that continuously |
| | taking the logarithm of a number, real or complex, will showcase a function |
| | that converges to the limit at a constant rate. Furthermore, collecting data |
| | using mathematical software supports the proof with hard data. Graphing |
| | the data points on the complex plane, we see that the function creates a |
| | spiral that has interesting properties on its own. With these new constants |
| | we can find new equilibriums in areas applying complex logarithms such |
| | as quantum mechanics, computer science, and electrical engineering. |
| | Thus, this research gives an interesting proof for a unique mathematical |
| | phenomenon in which potential real-world application may lie. |
| | Bibliography | Introduction to Complex Analysis by Rolf Nevanlinna and Veikko Paatero |
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