Question

Suppose $f\left(z\right)$ is a possibly complex valued function of a complex number z, which satisfies a function equation of the form $af\left(z\right)+bf\left(w^{2}z\right)=g\left(z\right)$ for all $zϵC,$ where a and b are some fixed complex numbers and $g\left(z\right)$ is some function of z and w is cube root of unity $(w\ne 1)$, then $f\left(z\right)$ can be determined uniquely if

• A. a + b = 0
• B. a² + b² ≠ 0
• C. a³ + b³ ≠ 0
• D. a³ + b³ = 0

Answer 1

The correct option is C a³ + b³ ≠ 0

Given $af\left(z\right)+bf\left(w^z\right)=g\left(z\right)...\left(i\right)$
Replace z by $wz$ and $w^{2}z,$ we get
$af\left(wz\right)+bf\left(z\right)=g\left(wz\right)....\left(ii\right)$
and $af\left(w^{2}z\right)+b\left(wz\right)=g\left(w^{2}z\right)....\left(iii\right)$
Equations (1),(2) and (3) together from a linear system of three equations in
three unknowns $f\left(z\right),f\left(wz\right)$ and $f\left(w^{2}z\right)$ which can be expressed in a matrix form as
$\left[\begin{array}{ccc}a& 0& b\\ b& a& 0\\ 0& b& a\end{array}\right]$
$\left[\begin{array}{c}f\left(z\right)\\ f\left(wz\right)\\ f\left(w^{2}z\right)\end{array}\right]$=$\left[\begin{array}{c}g\left(z\right)\\ g\left(wz\right)\\ g\left(w^{2}z\right)\end{array}\right]$
The determinant of the coefficient matrix is $a^3+b^3.$ So, when $a^3+b^3\ne 0$,the
functional equation (i) has a unique solution.Further $a^3+b^3\ne 0\Rightarrow a+b\ne 0.$