# Fundamental period of a function.

#### thumb2

Joined Oct 4, 2015
122
If we try to determine the fundamental period of the function:

$$4\cos(\pi t)\sin\left({\pi t \over 7}\right ) + 1$$

we have

\begin{align} & T_1 = {2\pi \over \pi} = 2\\ & T_2 = 2\pi /\left ({\pi \over 7} \right) = 14\\ & \mathrm {lcm}(2,14) = 14 \end{align}

(Latex here doesn't work very fine).

But if we graph the function, the period is 7 (see attachment).

My professor says that the period is 14.
I am convinced that is 7.

What's wrong with me ? Last edited:

#### thumb2

Joined Oct 4, 2015
122
Someone kindly corrected me that the lcm method is used with a sum of function and NOT with a product.
Expanding the function using complex exponentials, results in a sum of complex exponentials which fundamental period calculated with the lcm is 7.

Problem solved.

Last edited:

#### WBahn

Joined Mar 31, 2012
24,974
Have you shown that plot to your professor?

You first need to divide each by the greatest common divisor, which is 2. So you are looking for the LCM of 1 and 7, which is 7.

Remember, what you are really looking for are the smallest pair of values for which mT1 = nT2.

• thumb2

#### thumb2

Joined Oct 4, 2015
122
WBahn said:
Have you shown that plot to your professor?
Yes, two times, not one. TWO.

What makes me a little bit angry is:

1) Someone teach me wrong things. Everyone can make mistakes but I don't like when someone teach something wrong and it's convinced about it.
2) In front of the evidence, as the graph is, receiving replies as: "It's impossible that the period is 7" or other excuses useful just to hide some error.

Regards

• Monika Verma

#### WBahn

Joined Mar 31, 2012
24,974
Then your next step would be to show that T = 7s is an integer number of periods for that function.

Use the definition that T is a period of f(t) if and only if f(t+T) = f(t) for any value of T.

Crank the math and show that to your professor and tell him that you can't see where you have gone wrong and ask him to show you.

Make every step explicit and very easy to follow.

#### thumb2

Joined Oct 4, 2015
122
WBahn, I have used the standard definition too.

I checked it through Geogebra too, as it made me fall in doubt... Despite the graph, I was wrong.

Even a blind can see that the function is periodic with T = 7.

I definitely arrived at the mathematical proof that the lcm is true and valid only with sum of functions (even is sometimes the result is correct with product of functions). It can be easily demonstrated expanding the function to the complex exponential notation.