Feynmann's trick

Author: kevinlinxc

content/formulas/feynmanns-trick/index.md

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+---
+title: "Feynmann's Trick For Exponential Integrals"
+description: "A niche trick for evaluation exponential integrals of a certain form."
+summary: "A niche trick for evaluation exponential integrals of a certain form."
+tags: ["math", "calculus"]
+date: 2023-03-16
+latex: \int_0^{\infty} x^n e^{-tx} dx = \frac{n!}{t^{n+1}}
+---
+{{< katex >}}
+
+## Feynmann's Trick For Exponential Integrals
+$$\int_0^{\infty} x^n e^{-tx} dx = \frac{n!}{t^{n+1}}$$
+
+Where
+* \\(t\\) is a constant and
+* \\(n!\\) is the Gamma function (does not require \\(n\\) to be an integer)
+
+Feynmann's trick involves "differentiating under the integral" which is a whole technique on its own, and the above formula is something that can be found using this technique.
+I found that formula particularly hard to find online, so I decided to write it up.
+
+## Sources
+- [William's College (PDF Download)](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwi3t_De6eH9AhVzJH0KHZZRAqEQFnoECAoQAQ&url=https%3A%2F%2Fweb.williams.edu%2FMathematics%2Flg5%2FFeynman.pdf&usg=AOvVaw3M8KPQi7yOMURJvX58YdNj)