A functional equation

Aug 2018
137
7
România
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:
Dec 2015
1,082
169
Earth
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 \(\displaystyle f(1+i\sqrt{2} ) + f(1) +f(1-i\sqrt{2} )=f(-i\sqrt{2} )+f(0)+f(i\sqrt{2})\).
 
Aug 2018
137
7
România
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