- #1
utkarshakash
Gold Member
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Homework Statement
Suppose f(z) is a possibly complex valued function of a complex valued function of a complex number z, which satisfies a functional equation of the form [itex]af(z)+bf(\omega ^2 z)=g(z)[/itex] for all z in C, where a and b are some fixed complex numbers and g(z) is some function of z and ω is cube root of unity (ω≠1), then f(z) can be determined uniquely if
a)a+b=0
b)a^2+b^2≠0
c)a^3+b^3≠0
d)a^3+b^3=0
The Attempt at a Solution
If I substitute ω^2z in place of z the equation reduces to
[itex]af(\omega ^2 z) + bf(\omega z) = g(\omega ^2 z)[/itex]
Now if I add both eqns [itex]f(\omega ^2 z)(a+b)+af(z)+bf(\omega z)=g(z)+g(\omega ^2 z) [/itex]