Chapter 3: Q.14p (page 136)

In Problems $8 to 15, use (8.5)$ show that the given functions are linearly independent.
$14. e^{i x}, e^{- i x}$

Short Answer

Expert verified

It has been shown that the functions $e^{i x}, e^{- i x}$ are linearly independent for all values of $x$

Step by step solution

Step 1: Definition of linearly independent functions

The functions $f_{1} (x), f_{2} (x), \dots, f_{n} (x)$ are linearly independent if the determinant

localid="1657470388333" $W = |\begin{array}{cccc} f_{1} (x) & f_{2} (x) & \dots & f_{n} (x) \\ f_{1}^{'} (x) & f_{2}^{'} (x) & \dots & f_{n}^{'} (x) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ f_{1}^{(n - 1)} (x) & f_{2}^{(n - 1)} (x) & \dots & f_{n}^{(n - 1)} (x) \end{array}| \neq 0$

Here, W is called the Wronksian of functions.

Step 2: Use the Wronksian to show that the given functions are linearly independent.

Find the derivatives of the function $f_{1} (x) = e^{i x} of order 1$

$\begin{aligned} f_{1} (x) = e^{i x} \\ f_{1}^{'} (x) = i e^{i x} \end{aligned}$

Find the derivatives of the function $f_{2} (x) = e^{- i x} of order 1$

$\begin{aligned} f_{2} (x) = e^{- i x} \\ f_{2}^{'} (x) = - e^{- i x} \end{aligned}$

Substitute the derivatives in the Wronksian formula and simplify as follows:

$\begin{aligned} W = |\begin{array}{cc} e^{i x} & e^{- i x} \\ i e^{i x} & - i e^{- i x} \end{array}| \\ = (e^{i x} \times i e^{- i x}) - (e^{- i x} \times i e^{i x}) \\ = - i - i \\ = - 2 i \end{aligned}$

Since , $W \neq 0, therefore, e^{i x}, e^{- i x}$ are the linearly independent functions.

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In Problems $8 to 15, use (8.5)$ show that the given functions are linearly independent.
$14. e^{i x}, e^{- i x}$

Short Answer

Step by step solution

Step 1: Definition of linearly independent functions

Step 2: Use the Wronksian to show that the given functions are linearly independent.

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In Problems8to15,use(8.5) show that the given functions are linearly independent. 14.eix,e−ix

Short Answer

Step by step solution

Step 1: Definition of linearly independent functions

Step 2: Use the Wronksian to show that the given functions are linearly independent.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electrostatics

Magnetism

Force

Modern Physics

Work Energy and Power

Medical Physics

Study anywhere. Anytime. Across all devices.

In Problems $8 to 15, use (8.5)$ show that the given functions are linearly independent.
$14. e^{i x}, e^{- i x}$