Chapter 11: Q6P (page 549)

Assuming that x is real, show the following relation between the error function and the Fresnel integrals.
$e r f (\frac{1 - i}{\sqrt{2}} . x) = (1 - i) \sqrt{\frac{2}{π}} \int_{0}^{x} (c o s u^{2} + i s i n u^{2}) d u$

Short Answer

Expert verified

The relation between error function and Fresnel identity is established as $e r f (\frac{1 - i}{\sqrt{2}} . x) = (1 - i) \sqrt{\frac{2}{π}} \int_{0}^{x} (\cos u^{2} + i \sin u^{2}) d u$ .

Step by step solution

Given information

It is given that x is real in error function.

Formula of the error function

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

Establish the relation

Use the formula for the error function and substitute x by $\frac{1 - i}{\sqrt{2}}$ we get role="math" localid="1664350246689" $\frac{2}{\sqrt{π}} \int_{0}^{\frac{1 - i}{\sqrt{2}} . x} e^{- t^{2}} d t$ .

Then put $t = \frac{1 - i}{\sqrt{2}} u$ .

Which implies, $d t = \frac{1 - i}{\sqrt{2}} d u$ .

Integral becomes as $\frac{2}{\sqrt{π}} \int_{0}^{x} e^{- (\frac{{(1 - i)}^{2}}{2} . u^{2})} \frac{1 - i}{\sqrt{2}} d u$ .

$\begin{array}{rcl} {(1 - i)}^{2} & = & 1 - 2 i - 1 \\ {(1 - i)}^{2} & = & - 2 i \\ - (\frac{{(1 - i)}^{2}}{2}) & = & - \frac{- 2 i}{2} = i \end{array}$

Now the integral becomes as role="math" localid="1664350398123" width="148" height="67">1-i2π∫0xe-iu2du.

Use Euler identity

The integral after using Euler identity is mentioned below.

$(1 - i) \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- i u^{2}} d u = (1 - i) \frac{2}{\sqrt{π}} \int_{0}^{x} (\cos u^{2} + i \sin u^{2}) d u$

Hence proved.

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Assuming that x is real, show the following relation between the error function and the Fresnel integrals.
$e r f (\frac{1 - i}{\sqrt{2}} . x) = (1 - i) \sqrt{\frac{2}{π}} \int_{0}^{x} (c o s u^{2} + i s i n u^{2}) d u$

Short Answer

Step by step solution

Given information

Formula of the error function

Establish the relation

Use Euler identity

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Assuming that x is real, show the following relation between the error function and the Fresnel integrals.erf(1-i2.x)=(1-i)2π∫0x(cosu2+isinu2)du

Short Answer

Step by step solution

Given information

Formula of the error function

Establish the relation

Use Euler identity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Further Mechanics and Thermal Physics

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Study anywhere. Anytime. Across all devices.

Assuming that x is real, show the following relation between the error function and the Fresnel integrals.
$e r f (\frac{1 - i}{\sqrt{2}} . x) = (1 - i) \sqrt{\frac{2}{π}} \int_{0}^{x} (c o s u^{2} + i s i n u^{2}) d u$