Use the results of Problem 14 to find power series solution for the famous Gaussian integral below used in Physics, Astronomy, Chemistry, Statistics and many, many other places. This problem is also famous in that it must have an antiderivative because e-x² is continuous, however this is not an elementary function. e-x² dx e
Use the results of Problem 14 to find power series solution for the famous Gaussian integral below used in Physics, Astronomy, Chemistry, Statistics and many, many other places. This problem is also famous in that it must have an antiderivative because e-x² is continuous, however this is not an elementary function. e-x² dx e
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 14 result: Maclaurin Series for sinx = x - x3/3! + x5/5! - x7/7! + x9/9! - . . .
![### Gaussian Integral and Power Series Solution
Use the results of Problem 14 to find a power series solution for the famous Gaussian integral below, used in Physics, Astronomy, Chemistry, Statistics, and many other fields. This problem is also notable because, despite \( e^{-x^2} \) being continuous, it does not have an elementary antiderivative.
\[ \int e^{-x^2} \, dx \]
This integral is fundamental in various scientific fields due to Gaussian functions' prominence in representing normal distributions, wave functions, and other critical phenomena. The lack of an elementary antiderivative for \( e^{-x^2} \) makes this problem intriguing and motivates the exploration of power series solutions and other numerical methods to solve it.
For more detailed information on solving this integral using a power series and understanding its applications in different academic disciplines, refer to advanced calculus and mathematical analysis resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42eb3f21-115e-47f2-b59c-753292f0c3d6%2F84c0e5ec-d50e-46d3-b37c-0aa2267c80f5%2F0tee91_processed.png&w=3840&q=75)
Transcribed Image Text:### Gaussian Integral and Power Series Solution
Use the results of Problem 14 to find a power series solution for the famous Gaussian integral below, used in Physics, Astronomy, Chemistry, Statistics, and many other fields. This problem is also notable because, despite \( e^{-x^2} \) being continuous, it does not have an elementary antiderivative.
\[ \int e^{-x^2} \, dx \]
This integral is fundamental in various scientific fields due to Gaussian functions' prominence in representing normal distributions, wave functions, and other critical phenomena. The lack of an elementary antiderivative for \( e^{-x^2} \) makes this problem intriguing and motivates the exploration of power series solutions and other numerical methods to solve it.
For more detailed information on solving this integral using a power series and understanding its applications in different academic disciplines, refer to advanced calculus and mathematical analysis resources.
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