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This might be a very obvious one, but I am stuck on this from a long time.

If $F(s) = M(s) + N(s)$ where $M(s)$ is even polynomial function and $N(s)$ is odd polynomial function (where $s$ is a complex number), then how do I prove that

$${M(jw)}^2 - {N(jw)}^2 = {M(w)}^2 + {N(w)}^2$$

where $j = (-1)^{1/2}$ and $w$ is real?

I am also now really confused as to what exactly is the definition of odd and even functions in the complex domain. Kindly help.

As experts say, this seems to be incorrect or atleast incomplete. However, I have taken this from a standard textbook on Network analysis and synthesis by F. F. Kuo. If this helps someone in finding the appropriate conditions under which this is true, kindly help. The book doesn't seem to indicate anything besides those I have mentioned. Thank you for the responses so far.

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    The definition of odd and even functions is exactly the same. $f:\Bbb C\to\Bbb C$ is even if $f(-z)=f(z)$ for all $z$, and odd if $f(-z)=-f(z)$ for all $z$.2012-11-15
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    Do you have any assumptions on $M,N$ other than even/oddness? LHS depends only on values of $M,N$ at imaginary values and RHS on values of $M,N$ at real values. Doesn't seem to be true.2012-11-15
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    $M$,$N$ both are polynomials in $s$. Does that help?2012-11-15
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    I agree with sdcvvc: it sounds like you are leaving something out. I speculated that it was multiplicativity in my answer, since that more or less obviously leads to the solution.2012-11-15
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    @Guanidene Well yes that would certainly help...2012-11-15
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    @rschwieb Does your answer below fits for this condition? It doesn't seems to be clear from your answer. Kindly elaborate.2012-11-15
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    @Guanidene No, a polynomial function is not multiplicative, in general. But I think your problem seems to be false for polynomials if $M(w)=0$ and $N(w)=w$ for all $w$. I see now that another poster gave a second counterexample in polynomials, also.2012-11-15
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    @rschwieb Yes it seems to be false to me too but I am taking this from a standard Network analysis and synthesis textbook by f.f. kuo. Can't understand what aspect am I missing here. I'll try to let you know soon. Thanks for the help so far.2012-11-15
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    @rschwieb: $M(w)=0$ and $N(w)=w$ is not a counterexample.2012-11-15
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    @sdcvvc Doh you're right... somehow I had been cubing it.2012-11-15

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