Chapter 11: Q2P (page 548)

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 $t = (x), u = \frac{x}{\sqrt{2}}$ .

Short Answer

Expert verified

The equation has been proved.

Step by step solution

Given Information

The value of integration is $ϕ (x) = \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} e^{\frac{- t^{2}}{2}} d t$ .

Definition of error of function.

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

Find the value of Integral.

The value of integration is $ϕ (x) = \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} e^{\frac{- t^{2}}{2}} d t$ .

Substitute the value given below in the integration mentioned above.

$t = u \sqrt{2} d t = d v \sqrt{2}$

The equation becomes as follows.

$ϕ (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\frac{x}{\sqrt{2}}} e^{- u^{2}} \sqrt{2} d u ϕ (x) = \frac{1}{\sqrt{2 π}} [\int_{- \infty}^{0} e^{- u^{2}} \sqrt{2} d u + \int_{0}^{\frac{x}{\sqrt{2}}} e^{- u^{2}} \sqrt{2} d u]$

Separate the second term in the above equation.

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

Substitute the above value in role="math" localid="1664341673854" $\int_{- \infty}^{0} e^{- u^{2}} \sqrt{2} d u$ .

$\int_{- \infty}^{0} e^{- u^{2}} \sqrt{2} d u = - \int_{\infty}^{0} e^{- g^{2}} d g \int_{- \infty}^{0} e^{- u^{2}} \sqrt{2} d u = - \int_{0}^{\infty} e^{- g^{2}} d g \int_{- \infty}^{0} e^{- u^{2}} \sqrt{2} d u = \frac{\sqrt{π}}{2} e r f (\infty) \int_{- \infty}^{0} e^{- u^{2}} \sqrt{2} d u = \frac{\sqrt{π}}{2}$

Substitute the values in the equation mentioned below.

$ϕ (x) = \frac{1}{\sqrt{2 π}} [\int_{- \infty}^{0} e^{- u^{2}} \sqrt{2} d u + \int_{0}^{\frac{x}{\sqrt{2}}} e^{- u^{2}} \sqrt{2} d u] ϕ (x) = \frac{1}{\sqrt{π}} [\frac{\sqrt{π}}{2} + \frac{\sqrt{π}}{2} e r f [\frac{x}{\sqrt{2}}]] ϕ (x) = 0.5 + 0.5 e r f [\frac{x}{\sqrt{2}}]$

Hence, the solution is mentioned below.

The equation has been proved.

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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 $t = (x), u = \frac{x}{\sqrt{2}}$ .

Short Answer

Step by step solution

Given Information

Definition of error of function.

Find the value of Integral.

One App. One Place for Learning.

Most popular questions from this chapter

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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 theϕ(x) integral. Warning: Don’t forget to adjust the limits; whent=(x),u=x2 .

Short Answer

Step by step solution

Given Information

Definition of error of function.

Find the value of Integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Fundamentals of Physics

Fields in Physics

Quantum Physics

Conservation of Energy and Momentum

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

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 $t = (x), u = \frac{x}{\sqrt{2}}$ .