Can x! be Resolved in a Polynomial Form?

In summary, the conversation discusses the possibility of resolving the value of x! to a polynomial and integrating it normally. However, this is not possible as x! is only defined for non-negative integers and cannot be integrated. The conversation also mentions the Gamma function as a potential extension for x! and references the Sterling Series as a possible solution. The conversation also touches on the difficulty of integrating non-polynomial functions and clarifies that x! is a function from non-negative integers to non-negative integers.
  • #1
mathelord
I still think the x! can be resolved 2 a polynomial and then solved normally,or can't it,cos my friend is already working on it and has began to mke progress.
Or can't the x! be resolved let me know
 
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  • #2
What do you mean "..the x! can be resolved 2 a polynomial and then solved normally.."?

Factorial is only defined on non-negative integers, and not on any real interval (not considering a single point an interval here), so asking about it's integral is a bit of nonsense. Or do you mean factoria'ls usual extension to the Gamma function?
 
  • #3
mathelord said:
I still think the x! can be resolved 2 a polynomial and then solved normally,or can't it,cos my friend is already working on it and has began to mke progress.
Or can't the x! be resolved let me know
Almost no functions from Real numbers to Real numbers can be integrated in terms of elementary functions. To my knowledge only polynomials can be integrated to polynomials.

x! is not a function from real numbers to real numbers, it is a function from non-negative integers to non-negative integers and can't be integrated. Even if it could be integrated it grows faster than any polynomial so certainly its integral couldn't even be approximated by a polynomial over its whole domain.
 
  • #5
You could have spelled his name right...

Daniel.
 

FAQ: Can x! be Resolved in a Polynomial Form?

What is the integral of x?

The integral of x is a mathematical operation that calculates the area under the curve of a function. It is represented by the symbol ∫x and is also known as the antiderivative of x. The integral of x can be expressed as x^2/2 + C, where C is a constant.

How is the integral of x calculated?

The integral of x is calculated using a mathematical process called integration. This involves finding the antiderivative of x and then evaluating it at the upper and lower limits of the integration. The result is the difference between the two values, representing the area under the curve.

What is the significance of the integral of x in mathematics?

The integral of x is a fundamental concept in calculus and is used to solve many real-world problems. It is also important in physics and engineering, as it allows for the calculation of areas, volumes, and other quantities that change continuously.

Can the integral of x be negative?

Yes, the integral of x can be negative. This occurs when the function has negative values or when the upper limit of the integration is less than the lower limit. In this case, the result of the integral will be a negative value, representing the area below the x-axis.

What are some practical applications of the integral of x?

The integral of x has many practical applications, including calculating the area under a curve in economics, determining displacement in physics, and finding the volume of irregular objects in engineering. It is also used in statistics to calculate probabilities and in optimization problems to find the maximum or minimum value of a function.

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