Chapter 11: Q4P (page 548)

Show that $\int_{- \infty}^{\infty} e^{- \frac{y^{2}}{2}} d t = \sqrt{2 π}$
(a)by using (9.5) and (9.2a) .
(b) by reducing it to a $Γ$ function and using (5.3) .

Short Answer

Expert verified

By using (9.5) and (9.2a), it has been proved that $\int_{- \infty}^{\infty} e^{\frac{- y^{2}}{2}} d t = \sqrt{2 π}$ .
It has been proved that $\int_{- \infty}^{\infty} e^{\frac{- y^{2}}{2}} d t = \sqrt{2 π}$ .

Step by step solution

Given Information

The given statement to prove is $\int_{- \infty}^{\infty} e^{\frac{- y^{2}}{2}} d t = \sqrt{2 π}$ .

Definition of error of function

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

Prove the statement by theorems

(a)

The function $\sqrt{2 π} ϕ (\infty) = \int_{- \infty}^{\infty} e^{\frac{- u^{2}}{2}} d u$ .

Solve further to get the required answer.

$\begin{array}{rcl} \sqrt{2 π} ϕ (\infty) & = & \int_{- \infty}^{\infty} e^{\frac{- u^{2}}{2}} d u \\ \sqrt{2 π} [\frac{1}{2} + \frac{1}{2} e r f (\infty)] & = & \int_{- \infty}^{\infty} e^{\frac{- u^{2}}{2}} d u \\ \sqrt{2 π} & = & \int_{- \infty}^{\infty} e^{\frac{- u^{2}}{2}} d u \end{array}$

Or it can be said that role="math" localid="1664345288861" width="135" height="65">∫-∞∞e-y22dy=2π, hence proved.

Prove the statement by gamma theorem

Part (b)

Convert $\begin{array}{rcl} \int_{- \infty}^{\infty} e^{\frac{- u^{2}}{2}} d u \end{array}$ into gamma function to get the required results.

$\begin{array}{rcl} \int_{- \infty}^{\infty} e^{\frac{- u^{2}}{2}} d u & = & {2 \int}_{0}^{\infty} e^{\frac{- u^{2}}{2}} d u \end{array}$

Consider the following:

$s = \frac{u^{2}}{2} u = \sqrt{2 s} d u = \frac{1}{\sqrt{2 s}} d s$

Substitute the above value in $\begin{array}{rcl} \int_{0}^{\infty} e^{\frac{- u^{2}}{2}} d u \end{array}$ .

$\frac{2}{\sqrt{2}} \begin{array}{rcl} \int_{0}^{\infty} \end{array} \begin{array}{rcl} e \end{array} \begin{array}{rcl} \end{array}$ ^-ss^-12ds=2Γ12 22∫0∞e-ss-12ds=2Γ

Hence, $\sqrt{2 π} = \begin{array}{rcl} \int_{- \infty}^{\infty} \end{array} \begin{array}{rcl} e \end{array} \begin{array}{rcl} \end{array}$ ^-u22duor∫-∞∞e^-y22dy=2πequation has been proved.

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Show that $\int_{- \infty}^{\infty} e^{- \frac{y^{2}}{2}} d t = \sqrt{2 π}$
(a)by using (9.5) and (9.2a) .
(b) by reducing it to a $Γ$ function and using (5.3) .

Short Answer

Step by step solution

Given Information

Definition of error of function

Prove the statement by theorems

Prove the statement by gamma theorem

One App. One Place for Learning.

Most popular questions from this chapter

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Show that ∫-∞∞e-y22dt=2π(a)by using (9.5) and (9.2a) .(b) by reducing it to a Γ function and using (5.3) .

Short Answer

Step by step solution

Given Information

Definition of error of function

Prove the statement by theorems

Prove the statement by gamma theorem

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Rotational Dynamics

Oscillations

Measurements

Circular Motion and Gravitation

Modern Physics

Study anywhere. Anytime. Across all devices.

Show that $\int_{- \infty}^{\infty} e^{- \frac{y^{2}}{2}} d t = \sqrt{2 π}$
(a)by using (9.5) and (9.2a) .
(b) by reducing it to a $Γ$ function and using (5.3) .