# Solving Gamma Function Integral: e^(4u)*e^(-e^u)du

• splitendz
In summary: The symbols you are seeing are generated by Latex which is the preferred method of displaying math in this forum.In summary, the conversation is about solving the integral e^(4u) * e^(-e^u)du with upper limit infinity and lower limit 0. The suggested method is to substitute x = e^u and use the formula dx = e^u du. This simplifies the integral to \int \limits_1^\infty x^3 e^{-x} dx. The conversation also briefly discusses how to write sophisticated symbols using Latex formatting.
splitendz
Hi. I'm having some trouble solving the following gamma function:

Evaluate the integral e^(4u) * e^(-e^u)du. The upper limit is inifinity and the lower limit is 0.

I'm letting x = e^(u) or u = 1 in the hope to have the function looking similar to the gamma function. But I'm having no luck as du/dx will be equal to zero in this case.

You want

$$\int \limits_0^\infty e^{4u}e^{-e^u} du$$

Put $x=e^u$, so that $dx=e^u du$ and you get:

$$\int \limits_1^\infty x^3 e^{-x} dx$$

Can you go from there?

James R said:
You want

$$\int \limits_0^\infty e^{4u}e^{-e^u} du$$

Put $x=e^u$, so that $dx=e^u du$ and you get:

$$\int \limits_1^\infty x^3 e^{-x} dx$$

Can you go from there?

How do you write those sophisticated symbols?

I want to be able to write it, but don't know how

(Sorry its not helping your problem dude)

Thanks James. I'm right to continue now but shouldn't it be x^4 not x^3?

Actually, dx=e^u du, so one of the e^u's is in the dx, leaving only e^3u, or x^3.

Of course. Thanks for your help guys :) :)

I think part integrating will do it.Three times,i guess.

Daniel.

PhysicsinCalifornia said:
How do you write those sophisticated symbols?

I want to be able to write it, but don't know how

(Sorry its not helping your problem dude)

If you go to the "general physics" forum you will find a "sticky" on Latex formatting.

## 1. What is the gamma function?

The gamma function is a mathematical function that extends the factorial function to complex and real numbers, except for non-positive integers. It is denoted by the Greek letter Γ (gamma) and is defined as Γ(x) = ∫ e^(-t) t^(x-1) dt, where the integral is taken from 0 to ∞.

## 2. What is the purpose of solving the gamma function integral?

The gamma function integral is useful in many areas of mathematics, such as probability theory, number theory, and complex analysis. It also has applications in physics, particularly in the field of quantum mechanics.

## 3. How is the gamma function integral solved?

The gamma function integral can be solved using various techniques, such as integration by parts, substitution, and contour integration. The specific method used depends on the integrand and the limits of integration.

## 4. What is the significance of the expression e^(4u)*e^(-e^u) in the gamma function integral?

The expression e^(4u)*e^(-e^u) is the integrand of the gamma function integral and is commonly used in the evaluation of the integral. It is also related to the exponential function, which plays a crucial role in many mathematical and scientific applications.

## 5. Are there any real-world applications of solving the gamma function integral?

Yes, there are many real-world applications of the gamma function integral. For example, it is used in calculating the probability of rare events, estimating the number of species in a habitat, and analyzing the stability of a system in physics. It also has applications in engineering, finance, and statistics.

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