Evaluate some kind of gamma function

In summary, the conversation discusses two integrals, one of which makes no sense and the other is far easier to solve. The solution provided is to change the order of integration and the resulting integral is much easier to solve.
  • #1
ozgunozgur
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My question and solution that I've tried out are in attachment. Is it true my steps?
 

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  • #2
I am puzzled that you post the problem as $\int_0^1\int_{\sqrt{x}}^1e^{y^y} dxdy$, which makes no sense to me but do it as $\int_0^1\int_{\sqrt{x}}^1e^{y^3} dydx$ which does make sense and is far easier! Which is it?

The first integral makes no sense because the integral with respect to x has a function of x as the lower bound so that, even after the first integral, you will have a function of both x and y and after integrating with respect to y you will still have a function of x instead of a number.

And the second integral is far easier because $e^{y^y}$ is a horrendous function to integrate while $e^{y^3}$ is much easier!

To integrate $\int_0^1\int_{\sqrt{x}}^1e^{y^3} dydx$, I would first change the order of integration. The integral, taking x from 0 to 1 and, for each x, y from $\sqrt{x}$ to 1, is the portion of the square $0\le x\le 1$, $0\le y\le 1$, above the graph of $y= \sqrt{x}$. That is also the portion to the right of $x= y^2$ so the integral is $\int_0^1\int_0^{y^2} e^{y^3}dxdy$. There is no "x" in the integrand so the first integral just results in $\int_0^1 y^2e^{y^3}dy$ which is easy.
 
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FAQ: Evaluate some kind of gamma function

What is a gamma function?

A gamma function is a mathematical function that extends the concept of factorial to real and complex numbers. It is denoted by the Greek letter gamma (Γ) and is defined as Γ(z) = ∫0 xz-1e-xdx, where z is a complex number.

What is the purpose of a gamma function?

A gamma function is primarily used to solve problems involving the gamma distribution, which is a probability distribution that is commonly used in statistics and probability theory. It is also used in various fields of mathematics, such as number theory and complex analysis.

How is a gamma function evaluated?

A gamma function can be evaluated using various methods, such as numerical integration, series expansion, and recurrence relations. In some cases, it can be simplified to special functions, such as the beta function or the incomplete gamma function.

What are the applications of a gamma function?

The gamma function has numerous applications in mathematics, physics, engineering, and other fields. It is used to solve problems involving probability, statistics, differential equations, and special functions. It is also used in the computation of areas, volumes, and other geometric quantities.

Are there any limitations to the gamma function?

Yes, the gamma function has some limitations. It is not defined for negative integers and zero, and it has singularities at negative half-integers. It is also not defined for complex numbers with a negative real part. Additionally, the gamma function can be computationally expensive to evaluate for large values of its argument.

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