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if $f_1(x)$ and $f_2(x)$ are periodic functions with periods $T_1$ and $T_2 $ respectively then we have $h(x)=f_1(x)+f_2(x)$ has period as,=

$$ \begin{cases} \frac{1}{2}LCM (T_1,T_2), & \text{if $f_1(x)$ and $f_2(x)$ are complementary pairwise comparable even functions.} \\ LCM (T_1,T_2), & \text{otherwise} \end{cases}$$

What are complementary pairwise comparable even functions?

I'm not getting how to start this even.Need some hint.

Styles
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  • You are asking us what you're talking about, did I get that right? –  May 28 '17 at 07:20
  • @ProfessorVector:Actually,i don't understand the question completely,as i know what are complementary periodic functions but i'm not getting the complete meaning of the phrase "complementary pairwise comparable even functions".Moreover, i do not know the formal definition of complementary functions but know the examples which are complementary functions,For example sin(x) and cos(x) are complementary functions to each other. – Styles May 28 '17 at 07:35
  • It would be better to tell us what you understood. As it is, your question doesn't make sense, we don't even know kind of functions you mean. Real functions with real arguments? Then, LCM rarely makes sense. Integer functions with integer arguments? You don't say that. Why is this question interesting or important, in your opinion? –  May 28 '17 at 07:40
  • @ProfessorVector:I'm studying PROPERTIES OF PERIODIC FUNCTIONS.The formula in OP is given in the text book,but the proof is not there but i wanted to prove this.There is no mention about the nature of functions whether $f$ is a real or complex function.For convenience,we may consider $f$ to be a real function with real argument. – Styles May 28 '17 at 07:57
  • @ProfessorVector:https://math.stackexchange.com/questions/1424456/what-are-complementary-pair-wise-comparable-functions?rq=1 this may help in complete understanding of the problem. – Styles May 28 '17 at 07:59
  • What is the meaning of LCM, then? The likely interpretation as "least common multiple" makes sense only for integers, in general. –  May 28 '17 at 08:01
  • @ProfessorVector:What is LCM(2/3,5/7,11/5)? – Styles May 28 '17 at 08:03
  • What is LCM(1, $\pi$)? It's waste of time, I see. –  May 28 '17 at 08:05
  • @ProfessorVector:You're saying that LCM makes sense only for natural numbers – Styles May 28 '17 at 08:19
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    Actually, LCM makes sense for real numbers, provided their ratios are rational. Otherwise, the numbers have no common multiple, let alone a smallest one. – Harald Hanche-Olsen May 28 '17 at 09:04
  • Presumably, we are not necessarily talking about minimal periods here. Take, for example, $f_1(x)=\cos x+\cos 3x$ and $f_2(x)=-\cos x+\cos 3x$. Then $f_1$ and $f_2$ have period $2\pi$, while $f_1+f_2$ has period $\frac23\pi$. In particular, these functions can't be complementary pairwise comparable even functions, whatever that is. The sum still has period $2\pi$, however, even though that is not its minimal period. – Harald Hanche-Olsen May 28 '17 at 09:13

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