- #1
Another1
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Question
\[ \int dx_1dx_2...dx_d e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} = \frac{\pi ^{d/2}(d/r)!}{(d/2)!} \]
How can I derive this answer?
How can I integral this problem?
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- Thread starter Another1
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SUMMARY
The integral of the function \( e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} \) over the \( d \)-dimensional space is given by the formula \( \frac{\pi^{d/2}(d/r)!}{(d/2)!} \). This result can be derived using techniques from multivariable calculus and properties of Gaussian integrals. Specifically, the derivation involves transforming the integral into polar coordinates and applying the Gamma function. Understanding these concepts is essential for accurately deriving the stated formula.
PREREQUISITES- Multivariable calculus
- Gaussian integrals
- Polar coordinates transformation
- Gamma function properties
- Study the derivation of Gaussian integrals in multiple dimensions
- Learn about the properties and applications of the Gamma function
- Explore polar coordinates transformations in higher dimensions
- Investigate advanced topics in multivariable calculus
Mathematicians, physicists, and students studying advanced calculus or mathematical analysis who are interested in integral calculus and its applications in higher dimensions.
Question
\[ \int dx_1dx_2...dx_d e^{(x^2_1+x^2_2+...+x^2_d)^{r/2}} = \frac{\pi ^{d/2}(d/r)!}{(d/2)!} \]
How can I derive this answer?
- #2
HOI
- 921
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"Integral" is a noun. The verb is "integrate".
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