# Integral function of order alpha

• jostpuur
In summary, an integral function of order \alpha is a complex analytic function \mathbb{C}\to\mathbb{C} that is also analytic on the punctured complex plane, \mathbb{C}\backslash\{z_1,\ldots, z_n\}\to\mathbb{C}. The order of an integral function refers to the power of the highest derivative in its Taylor series expansion. This term is synonymous with "entire function".
jostpuur
What does it mean when a function is said to be an integral function of order $\alpha$? $\alpha$ is some real constant. I encountered this terminology in the context of complex analytic functions.

I just realized that a complex function being integral very probably means that it is analytic function $\mathbb{C}\to\mathbb{C}$, and in particular not only an analytic function $\mathbb{C}\backslash\{z_1,\ldots, z_n\}\to\mathbb{C}$. I already knew a name for these functions in Finnish, but didn't realize immediately that this is how it gets translated into English.

I'm still wondering about what the order of an integral function means.

Thanks, that appears to be precisely what I was after. The text I was reading deals with similar expressions when talking about order.

Looking at the Mathworld, I noticed that I would have recognized the term "entire function", but did not know that "integral function" is synonymous with it.

## 1. What is the definition of the integral function of order alpha?

The integral function of order alpha is a mathematical function that is defined as the integral of a given function raised to the power of alpha, where alpha is a real number. It is denoted by ∫ab f(x)^α dx.

## 2. How is the integral function of order alpha different from the regular integral?

The integral function of order alpha is different from the regular integral in that it involves raising the function to a power of alpha before integrating. This means that the resulting integral function will have a different form and may have different properties compared to the regular integral.

## 3. What are some applications of the integral function of order alpha?

The integral function of order alpha has various applications in mathematical analysis, physics, and engineering. It is commonly used in solving differential equations, finding areas and volumes of irregular shapes, and calculating moments of inertia.

## 4. How do you calculate the integral function of order alpha?

The integral function of order alpha can be calculated using different integration techniques such as substitution, integration by parts, and partial fractions. The specific method used will depend on the form of the function being integrated and the value of alpha.

## 5. Is there a relationship between the integral function of order alpha and the gamma function?

Yes, there is a relationship between the integral function of order alpha and the gamma function. The gamma function is a generalization of the factorial function and can be expressed as a special case of the integral function of order alpha. This relationship is expressed as γ(x) = ∫0 e^-t t^(x-1) dt, where x > 0.

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