Chapter 2: Q18P (page 66)

Show that $[A e^{i k x} + B e^{- i k x}]$ and $[C c o s (k x) + D s i n (k x)]$ are equivalent ways of writing the same function of $x$ , and determine the constants $C$ and $D$ in terms of $A$ and $B$ , and vice versa.

Short Answer

Expert verified

The constants $C$ and $D$ in terms of $A$ and $B$ , and vice versa are,

(i) $C = A + B$ and $D = i (A - B)$

(ii) $A = \frac{1}{2} (C - i D); B = \frac{1}{2} (C + i D)$

Step by step solution

Step 1:Formulae used

The required formula are

$e^{ikx} = (c o s (k x) + i s i n (k x))$

$e^{- i k x} = (c o s (k x) - i s i n (k x))$

$c o s (k x) = (\frac{e^{i k x} + e^{- i k x}}{2})$

$s i n (k x) = D (\frac{e^{i k x} - e^{- i k x}}{2})$

Compute C and D in terms of A and B and vice versa

Apply Euler's formula,

$A e^{i k x} + B e^{- i k x} = A (\cos (k x) + i \sin (k x)) + B (\cos (k x) - i \sin (k x))$

$= (A + B) \cos (k x) + i (A - B) \sin (k x)$

$= C \cos (k x) + D \sin (k x)$

Thus, $C$ and $D$ expressed in terms of arbitrary constants $A$ and $B$ , $C = A + B$

$D = i (A - B)$

Now, Solve both the equations for $A$ and B and get,

$C \cos (k x) + D \sin (k x) = C (\frac{e^{i k x} + e^{i k x}}{2}) + D (\frac{e^{i k x} - e^{k x}}{2 i})$

$= \frac{1}{2} (C - i D) e^{i k x} + \frac{1}{2} (C + i D) e^{i k x}$

$= A e^{A x} + B e^{i x x}$

Therefore, the values of $A$ and $B$ are,

A = \frac{1}{2} (C - i D)

B = \frac{1}{2} (C + i D)

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Show that $[A e^{i k x} + B e^{- i k x}]$ and $[C c o s (k x) + D s i n (k x)]$ are equivalent ways of writing the same function of $x$ , and determine the constants $C$ and $D$ in terms of $A$ and $B$ , and vice versa.

Short Answer

Step by step solution

Step 1:Formulae used

Compute C and D in terms of A and B and vice versa

One App. One Place for Learning.

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Show that [Aeikx+Be-ikx] and [Ccos(kx)+Dsin(kx)] are equivalent ways of writing the same function of x, and determine the constants C and D in terms of Aand B, and vice versa.

Short Answer

Step by step solution

Step 1:Formulae used

Compute C and D in terms of A and B and vice versa

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

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Study anywhere. Anytime. Across all devices.

Show that $[A e^{i k x} + B e^{- i k x}]$ and $[C c o s (k x) + D s i n (k x)]$ are equivalent ways of writing the same function of $x$ , and determine the constants $C$ and $D$ in terms of $A$ and $B$ , and vice versa.