Question about Elementary Functions

In summary, the conversation discusses the concept of elementary functions and their integrals. It is noted that most elementary functions do not have elementary functions as integrals. The question is then raised about the smallest set of functions that includes elementary functions and is closed under integration. The definition of an elementary function is also discussed, including its components and operations. Finally, the complexity of describing integrals of all elementary functions is mentioned.
  • #1
lugita15
1,554
15
Obviously, most elementary functions do not have elementary functions as integrals. For instance, the integral of e^-(x^2) is not an elementary function even though its integrand is. My question is, what is the smallest set of functions which includes the set of elementary function and is closed under integration?
 
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  • #2
To some extent it depends on a precise definition of "elementary function". For example, the integral of e^[(-x^2)/2] (with appropriate normalizations) is called erf(x). Whether or not it is elementary becomes a matter of definition.
 
  • #3
As to what constitutes an elementary function I define it as follows:
An elementary function is a function built from a finite number of exponentials, logarithms, trigonometric functions, inverse trigonometric functions, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ - × ÷). The roots of equations are the functions implicitly defined as solving a polynomial equation with constant coefficients.
 
  • #4
even the simpler question of describing integrals of all elementary functions seems a tall order. i.e. of giving a collectioncontaining integrals of all elementarty functions and possibly not closed under integration.

i have no idea.

even the set of all elementary functions is a field extension of infinite transcendence degree over C(X) hence more than a bit complicated.
 

FAQ: Question about Elementary Functions

What are elementary functions?

Elementary functions are the basic mathematical functions that are commonly used in elementary algebra, such as addition, subtraction, multiplication, division, and exponentiation. They also include trigonometric, logarithmic, and exponential functions.

What is the difference between elementary and non-elementary functions?

The main difference between elementary and non-elementary functions is that elementary functions can be expressed using a finite number of operations and constants, while non-elementary functions require an infinite number of operations and constants to be expressed.

How are elementary functions used in science?

Elementary functions are essential in scientific calculations and modeling. They are used to describe and analyze physical phenomena, and are also used in statistics, engineering, and computer science.

Can elementary functions be graphed?

Yes, elementary functions can be graphed to visually represent their behavior and relationships. This is a useful tool for understanding and analyzing their properties.

What are some real-life applications of elementary functions?

Elementary functions have a wide range of real-life applications, such as calculating interest rates in finance, predicting natural phenomena in weather forecasting, and analyzing data in scientific research. They are also used in everyday tasks like measuring distance and time, and in creating computer algorithms.

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