Chapter 7: Q36E (page 281)

Prove that if the function $e^{i \sqrt{D} 蠁}$ is to meet itself smoothly when $蠁$ changes by $2 蟺$ , $\sqrt{D}$ must be an integer.

Short Answer

Expert verified

For the function $e^{i \sqrt{D} 蠁}$ to have a period of $2 蟺$ , $\sqrt{D}$ must be an integer.

Step by step solution

A concept:

The functions which repeat itself after regular interval of time is called a Cyclic function and that interval in which it is repeating itself is called the period of that cyclic function.

As you know that, by Euler鈥檚 equation,

$e^{ix} = \cos (x) + i \sin (x)$

Value of the function at φ=0 and φ=2π :

Let, the function be

$F (蠁) = e^{i \sqrt{D} 蠁} = \cos (\sqrt{D} 蠁) + isin (\sqrt{D} 蠁)$

Where, $D = - \frac{1}{桅} \frac{{鈭�}^{2} 桅}{鈭� 蠁^{2}}$ and $蠁$ is the Azimuthal Angle.

Now at $蠁 = 0$ :

$F (0) = e^{i \sqrt{D} 0} = \cos (\sqrt{D} 0) + isin (\sqrt{D} 0) = 1$

Again, if the function meets itself at $蠁 = 2 蟺$ ,

$F (2 蟺) = F (0) = 1$

Value of :

If, $F (2 蟺) = 1$

$e^{i \sqrt{D} 2 蟺} = 1 \cos (\sqrt{D} 2 蟺) + isin (\sqrt{D} 2 蟺) = 1$ 鈥�.. (1)

In equation (1), if the real part is 1 the imaginary part should be zero and for that to hold, must be an integer $\sqrt{D}$

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91影视

Prove that if the function $e^{i \sqrt{D} 蠁}$ is to meet itself smoothly when $蠁$ changes by $2 蟺$ , $\sqrt{D}$ must be an integer.

Short Answer

Step by step solution

A concept:

Value of the function at φ=0 and φ=2π :

Value of :

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91影视

Prove that if the functioneiD蠁is to meet itself smoothly when蠁changes by 2蟺, D must be an integer.

Short Answer

Step by step solution

A concept:

Value of the function at φ=0 and φ=2π :

Value of :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Energy Physics

Classical Mechanics

Waves Physics

Astrophysics

Electromagnetism

Study anywhere. Anytime. Across all devices.

Prove that if the function $e^{i \sqrt{D} 蠁}$ is to meet itself smoothly when $蠁$ changes by $2 蟺$ , $\sqrt{D}$ must be an integer.