How Can You Prove the Error Function in Mathematics?

[vtnsx]

Here's the question... i really don't know how to prove this question
http://members.shaw.ca/brian103/theerrorfunction.jpg

 

[Astronuc]

a)

\int_a^b f(x)\,dx = \int_0^b f(x)\,dx - \int_0^a f(x)\,dx

b) I think you just subsitute in the function y and y' and show the equation is satisfied.

 

[M98Ranger]

The error function is a mathematical function used in statistics and physics to represent the cumulative distribution function of a normal distribution. It is defined as:

erf(x) = (2/√π)∫e^(-t^2)dt from 0 to x

To prove this function, we can use the definition of the error function and the properties of integrals.

First, we can rewrite the integral as:

erf(x) = (2/√π)∫e^(-t^2)dt from -∞ to x

Next, we can use the substitution u = -t^2 and du = -2tdt to rewrite the integral as:

erf(x) = (-1/√π)∫e^u du from -∞ to -x^2

Using the fundamental theorem of calculus, we can evaluate the integral to get:

erf(x) = (-1/√π)(e^-x^2 - e^-∞)

Since e^-∞ is equal to 0, we can simplify the equation to:

erf(x) = (2/√π)e^-x^2

This is the same equation as the one given in the definition of the error function. Therefore, we have proven the error function.