A functional equation

Integrator

Hello all,

Resolve the functional equation $$\displaystyle f(x^n-i\sqrt2)+f(x^n)+f(x^n+i\sqrt2)=a+bi$$ where $$\displaystyle i^2=-1$$ , $$\displaystyle n\in \mathbb N$$* and $$\displaystyle a,b\in \mathbb R$$.

All the best,

Integrator

Last edited:

idontknow

Try to set x=1 and x=0 .
for x=1 , $$\displaystyle f(1+i\sqrt{2} ) + f(1) +f(1-i\sqrt{2} )=a+bi$$.
for x=0 , $$\displaystyle f(-i\sqrt{2} )+f(0)+f(i\sqrt{2})=a+bi$$.
Also $$\displaystyle f(1+i\sqrt{2} ) + f(1) +f(1-i\sqrt{2} )=f(-i\sqrt{2} )+f(0)+f(i\sqrt{2})$$.

Integrator

Try to set x=1 and x=0 .
for x=1 , $$\displaystyle f(1+i\sqrt{2} ) + f(1) +f(1-i\sqrt{2} )=a+bi$$.
for x=0 , $$\displaystyle f(-i\sqrt{2} )+f(0)+f(i\sqrt{2})=a+bi$$.
Also $$\displaystyle f(1+i\sqrt{2} ) + f(1) +f(1-i\sqrt{2} )=f(-i\sqrt{2} )+f(0)+f(i\sqrt{2})$$.
Hello,

I do not understand!What is the general form of the functions $$\displaystyle f(x)$$?
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For example, resolve the following case:
If $$\displaystyle n=2$$, $$\displaystyle a=2$$ and $$\displaystyle b=1$$ then what is the general form of the function $$\displaystyle f(x)$$?

All the best,

Integrator