Chapter 7: Q7-13-18MP (page 388)

Let f(x)and $g (α)$ be a pair of Fourier transforms. Show that $\frac{df}{dx}$ and $i α g (α)$ are a pair of Fourier transforms. Hint: Differentiate the first integral in (12.2)under the integral sign with respect to x. Use to show that
$\int_{- \infty}^{\infty} α {| g (α) |}^{2} d α = \frac{1}{2 π i} \int_{- \infty}^{\infty} \bar{f} (x) \frac{d}{d x} f (x) d x$ .

Short Answer

Expert verified

The Derivative of with respect to is $\frac{df}{dx} = \int_{- \infty}^{\infty} g (α) i α e^{i α x} d α$ $\int_{- \infty}^{\infty} α {|g (α)|}^{2} d α = \frac{1}{2 πi} \int_{- \infty}^{\infty} \bar{f} (x) \frac{d}{d x} f (x) d x$ is verified.

Step by step solution

Given information

We have given f(x) and $g (α)$ be a pair of Fourier transforms.

Definition of Inverse Fourier transformation

For a given function u(x) define its Inverse Fourier transformation U as

$U (k) {=f}^{- 1} {u (x)} = \frac{1}{\sqrt{2 π}} ∫_{- \infty}^{\infty} e^{i k x} u (x) dx$

Evaluate dfdx

By the definition of the inverse Fourier transformation

$f (x) {=∫}_{- \infty}^{\infty} g (α) e^{i k x} dα$

Differentiate it with respect to x

$\frac{d f}{d x} = \int_{- \infty}^{\infty} g (α) i α e^{i α x} d α$

Show that ∫-∞∞α|gα|2dα=12πi∫-∞∞f-(x)ddxf(x)dx

To show this, let’s start with equation 12.23

$\int_{- \infty}^{\infty} \bar{g_{1}} (α) \times g_{2} (α) d α = \frac{1}{2 π} \int_{- \infty}^{\infty} \bar{f_{1}} (x) \times f_{2} (x) d x$

Let $\bar{g_{1}} (α) = g, g_{2} (α) = i α g, \bar{f_{1}} (x) = \bar{f}$ and $f_{2} (x) = \frac{d f}{d x}$

Then result follows

$\int_{- \infty}^{\infty} α {|g (α)|}^{2} d α = \frac{1}{2 π i} \int_{- \infty}^{\infty} \bar{f} (x) \frac{d}{d x} f (x) d x$

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Short Answer

Step by step solution

Given information

Definition of Inverse Fourier transformation

Evaluate dfdx

Show that ∫-∞∞α|gα|2dα=12πi∫-∞∞f-(x)ddxf(x)dx

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Let f(x)andg(α)be a pair of Fourier transforms. Show thatdfdxandiαg(α)are a pair of Fourier transforms. Hint: Differentiate the first integral in (12.2)under the integral sign with respect to x. Use to show that∫-∞∞α|gα|2dα=12πi∫-∞∞f-(x)ddxf(x)dx.

Short Answer

Step by step solution

Given information

Definition of Inverse Fourier transformation

Evaluate dfdx

Show that ∫-∞∞α|gα|2dα=12πi∫-∞∞f-(x)ddxf(x)dx

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity and Magnetism

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Further Mechanics and Thermal Physics

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Study anywhere. Anytime. Across all devices.