# Show these functions are 2 pi periodic

• bbq pizza
In summary: I'm done.In summary, the conversation discussed a problem involving the functions g(t) and h(t) and their periodicity. It was stated that g(t) is equal to half of the sum of f(t) and f(-t), while h(t) is equal to half of the difference of f(t) and f(-t). The problem also involved showing that g(t+2π) and h(t+2π) are equivalent to g(t) and h(t), respectively. Further discussion was had about the role of -t in this problem and the need for f(t) to have a period of 2π for the problem to hold true. An example was provided using the exponential definition of the cosine

#### bbq pizza

Member warned about posting without the HW template
g(t)=½( f(t)+f(-t) ) h(t)=½( f(t)-f(-t) )
show its 2π periodic so: g(t+2π) = ½( f(t+2π)+f(t-2π) ) why does -t become t-2π ?
½( f(t)+f(-t) ) = g(t)
h(t+2π)=½( f(t+2π)-f(t-2π) )
½( f(t)-f(-t) ) = h(t) is this correct?
can anybody show me some similar examples please?
this is from a Fourier series question paper.
thanks

Of course, this problem is false unless ##f(t) = f(t+2\pi)##, which you haven't told us.

bbq pizza said:
g(t)=½( f(t)+f(-t) ) h(t)=½( f(t)-f(-t) )
show its 2π periodic so: g(t+2π) = ½( f(t+2π)+f(t-2π) ) why does -t become t-2π ?

It doesn't. If you put ##t+2\pi## in for ##t## in ##f(-t)## you get ##f(-(t+2\pi)) = f(-t -2\pi)##. Do you see why the numerator becomes ##f(t)+f(-t)##?

Delta2
An example would be the exponential definition of the cosine function.
##\cos(x) = \frac{1}{2} (e^{ix} + e^{-ix}) ##
This is analogous to your problem for ##g(x)## with ##f(x) = e^{ix}##.
##\cos(x+2\pi) = \frac{1}{2} (e^{i(x+2\pi)} + e^{-i(x+2\pi)})\\
= \frac{1}{2} (e^{ix}e^{i2\pi} +e^{-ix}e^{-i2\pi})\\
= \frac{1}{2} (e^{ix} + e^{-ix})\\
= \cos(x) ##

Hint: prove that if a function has period T, then ##f(x-T)=f(x)## as long as f is defined in x-T.

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LCKurtz said:
Of course, this problem is false unless ##f(t) = f(t+2\pi)##,
or maybe the sum of a periodic function and an odd function.

LCKurtz said:
Of course, this problem is false unless ##f(t) = f(t+2\pi)##, which you haven't told us.

haruspex said:
or maybe the sum of a periodic function and an odd function.

Well, I assumed, which was also unstated, that problem was to prove that if ##f## has period ##2\pi##, then when you write ##f = g + h## with ##g## even and ##h## odd, that ##g## and ##h## have period ##2\pi##. In that context I'm not sure what you are suggesting.

And, annoyingly enough and a pet peeve of mine, the OP hasn't returned to the thread to clarify anything

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## 1. What does it mean for a function to be 2 pi periodic?

Being 2 pi periodic means that the function repeats itself every 2 pi radians. This means that the behavior of the function will be the same after every 2 pi radians, creating a pattern that repeats infinitely.

## 2. How can you show that a function is 2 pi periodic?

To show that a function is 2 pi periodic, you can use the definition of a periodic function which states that f(x + T) = f(x) for all values of x, where T is the period. In this case, T = 2 pi, so you would need to show that f(x + 2 pi) = f(x) for all values of x.

## 3. Can any function be 2 pi periodic?

No, not all functions are 2 pi periodic. Only functions that exhibit a repeating pattern every 2 pi radians can be considered 2 pi periodic. For example, linear functions and exponential functions are not 2 pi periodic.

## 4. What are some common examples of 2 pi periodic functions?

Some common examples of 2 pi periodic functions include sine, cosine, tangent, and their inverse functions. These trigonometric functions have a repeating pattern every 2 pi radians, making them 2 pi periodic.

## 5. Why is it important to show that a function is 2 pi periodic?

Showing that a function is 2 pi periodic can be useful in many applications, particularly in physics and engineering. It allows us to predict the behavior of the function over a wide range of values, and simplifies calculations by reducing the domain of the function to just one period. Additionally, many real-world phenomena exhibit periodic behavior, so understanding 2 pi periodic functions can help us model and analyze these phenomena.