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Chapter 11: Q3P (page 548)

Prove that erf(x) is an odd function of x. Hint: Put t = -s in (9.1) .

Short Answer

Expert verified

It has been proved that the erf(x) is an off function of x .

Step by step solution

01

Definition of error of function

The error of the function is defined as $(x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t$ .

02

Prove that error is odd

The value of integration is $e r f (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t$ .

Substitute -x in place of x .

$e r f (- x) = \frac{2}{\sqrt{π}} \int_{0}^{- x} e^{- t^{2}} d t$

Let u = -s , then du = -ds .

The equation becomes as follows:

$e r f (- x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- s^{2}} (- d s) e r f (- x) = - e r f (x)$

Hence, the statement has been proved.

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Most popular questions from this chapter

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

8. $\int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

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Verify equations (9.2), (9.3), and (9.4). Hint: In (9.2a) , you want to write $ϕ (x)$ in terms of an error function. Make the change of variable t = u √2 t = u (√2)in the $ϕ (x)$ integral. Warning: Don’t forget to adjust the limits; when $t = (x), u = \frac{x}{\sqrt{2}}$ .

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