Question

A functionf(θ) is periodic if after some periodt, it repeats. In other wordsf(θ t) =f(θ) for allθ. Lettingθbe a real number, isf(θ) =eiθperiodic? (2 points) if so, what is its period? (2points)

Answer 1

The period of the given function is T = 2π

What is the period of the function?

The period T of a function f(x) is such that:

f(x + T) = f(x).

In this case, our function is:

f(θ) = e^{iθ}

Remember that this can be written as:

f(θ) = cos(θ) + i*sin(θ)

So yes, this is in did a periodic function.

Then the period of the function f(θ) is the same as the period of the cosine and sine functions, which we know is T = 2π.

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https://brainly.com/question/26449711

Answer 2

The period of the considered function f(θ) = e^{iθ} is found to be P = 2π (assuming 'i' refers to 'iota' and 'e' refers to the base of the natural logarithm)

What is euler's formula?

For any real value θ, we have:

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

where 'e' is the base of the natural logarithm, and 'i' is iota, the imaginary unit.

What are periodic functions?

Functions which repeats their values after a fixed interval, are called periodic function.

For a function $y = f(x)$, it is called periodic with period 'T' if we have:

$y = f(x) = f(x + T) \: \forall x \in D(f)$

where D(F) is the domain of the function f.

Suppose that, the period of the function $f(\theta) = e^{i \theta}$ be P, then we get:
$f(\theta + P) = f(\theta)\\\\e^{i(\theta)} = e^{i(\theta + P)}\\\\\cos(\theta) + i\sin(\theta) = \cos(\theta + P) + i\sin(\theta + P)$

When two complex numbers are equal, then their real parts are equal and their imaginary parts are equal.

That means,

$\cos(\theta) + i\sin(\theta) = \cos(\theta + P) + i\sin(\theta + P)$ implies that:

$\cos(\theta) = \cos(\theta + P)\\\sin(\theta) = \sin(\theta + P)$

Also, we know that the period of sine and cosine function is $2\pi$

Thus, we get:

$P = 2\pi$

Thus, the period of the function $f(\theta) = e^{i \theta}$  is P = 2π

Learn more about periodic functions here:

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