Integrating the Exponential Function: Techniques and Resources

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In summary, the integral can be expressed using the "special" function, \mbox{erf} (x), which is tabulated for real arguments only.
  • #1
metrictensor
117
1
Does anyone know how to go about solving

[itex]\int e^{(x^{2})} dx[/itex]
 
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  • #2
not expressible in terms of elementary functions. Using the "special" function, [itex]\mbox{erf} (x)[/itex] (the "error function", defined by [itex]\mbox{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2} \ dt[/itex]), you can express the integral in this way, though:

[tex]\int e^{(x^2)} \ dx = -\frac{i\sqrt{\pi}}{2}\mbox{erf}(ix)+C.[/tex]

Essentially what this means is that you either have to compute definite integrals with this integrand numerically, or look up values on tables for the error function.
 
  • #3
Data said:
not expressible in terms of elementary functions. Using the "special" function, [itex]\mbox{erf} (x)[/itex] (the "error function", defined by [itex]\mbox{erf}(x) = 2/\sqrt{\pi} \int_0^x e^{-t^2} \ dt[/itex]), you can express the integral in this way, though:

[tex]\int e^{(x^2)} \ dx = -\frac{i\sqrt{\pi}}{2}\mbox{erf}(ix)+C.[/tex]

Essentially what this means is that you either have to compute definite integrals with this integrand numerically, or look up values on tables for the error function.
This is helpful. I have the bounds. Thaks. It is weird that there is no analytical solution to something that looks so simple. I did write a taylor series and integrated that to get an infinite sum that is equal to the integral. It is:

[itex]
\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)n!}}
[/itex]
 
  • #4
that's like the classic example of something that has no elementary solution...
 
  • #5
The "erf" function is tabulated for real arguments only...


Daniel.
 
  • #6
Really? I have a tabulation for complex arguments in front of me right now!



:-p
 
  • #7
Give me a link to the page in A & Stegun where the erf function of complex arg is tabulated.

Daniel.
 

Related to Integrating the Exponential Function: Techniques and Resources

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to solve problems involving continuous functions and is an important tool in calculus.

2. How do you evaluate an integral?

To evaluate an integral, you can use different techniques such as substitution, integration by parts, and trigonometric substitution. You can also use tables, graphs, and numerical methods to approximate the value of an integral.

3. What is the purpose of evaluating an integral?

Evaluating an integral allows us to find the exact value of the area under a curve, which can be useful in many real-world applications. It also helps in solving problems related to velocity, acceleration, and other physical quantities.

4. What are the different types of integrals?

The two main types of integrals are indefinite and definite integrals. Indefinite integrals do not have limits of integration and result in a function, while definite integrals have specific limits of integration and result in a numerical value.

5. What is the relationship between integrals and derivatives?

The fundamental theorem of calculus states that the derivative of an integral is equal to the original function. This means that integrals and derivatives are inverse operations of each other, and they are closely related in calculus.

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