Gamma
Function
Independent Research Project
Introduction
The factorial of a number is that number multiplied by all the positive integers below it in value and has the form: n! = 1(2)...(n-1)(n) where n is equal to 0 or any arbitrary positive integer. The factorial function is an essential part of mathematics and statistics, yet it is defined for only 0 and any positive integer. This paper delves into the history, definition, properties, proof, and some applications of the Gamma function, which is able to find the factorial of other numbers, such as complex and rational numbers.
My Role
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Gather background information about the Gamma function, including:
history, definitions, properties, and applications -
Clearly explain an induction proof that proves two definitions of the Gamma function are equivalent
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Write code that graphs both the factorial and Gamma function together
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Provide examples of applying Analytic Continuation to the Gamma function
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Produce research paper through writing and figure creation
Technologies and Skills
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Matlab
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Utilize build-in Matlab functions such as gamma and factorial
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Overlay a scatterplot on top of a line graph and format the graph
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Mathematical Knowledge In:
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Linear Algebra
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Advanced Algebra
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Calculus
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Complex Analysis
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​Microsoft PowerPoint
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Microsoft Word ( + importing LaTeX syntax)
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Google Docs
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Google Drive
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Google Slides
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Zoom (remote communication)
Research Paper
Click the button to view the paper produced from the project.
Figure
The figure was constructed by importing the Matlab graph created through the Matlab code in the Appendix.
Graph of the factorial function and the Gamma function from n=0 to n=3. The blue points represent the factorial function, while the red line represents the Gamma function. The x-axis represents the value of n for the factorial function. The y-axis represents the factorial of n. To form a continuous graph, the Gamma function is shifted one unit to the left. The Matlab code used to produce the graph is in the Appendix of the paper.