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May 10th, 2019, 08:18 AM  #1 
Member Joined: Aug 2018 From: România Posts: 84 Thanks: 6  A functional equation
Hello all, Resolve the functional equation $\displaystyle f(x^ni\sqrt2)+f(x^n)+f(x^n+i\sqrt2)=a+bi$ where $\displaystyle i^2=1$ , $\displaystyle n\in \mathbb N$* and $\displaystyle a,b\in \mathbb R$. All the best, Integrator Last edited by Integrator; May 10th, 2019 at 08:22 AM. 
May 10th, 2019, 08:03 PM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 596 Thanks: 87 
Try to set x=1 and x=0 . for x=1 , $\displaystyle f(1+i\sqrt{2} ) + f(1) +f(1i\sqrt{2} )=a+bi$. for x=0 , $\displaystyle f(i\sqrt{2} )+f(0)+f(i\sqrt{2})=a+bi$. Also $\displaystyle f(1+i\sqrt{2} ) + f(1) +f(1i\sqrt{2} )=f(i\sqrt{2} )+f(0)+f(i\sqrt{2})$. 
May 10th, 2019, 08:53 PM  #3  
Member Joined: Aug 2018 From: România Posts: 84 Thanks: 6  Quote:
I do not understand!What is the general form of the functions $\displaystyle f(x)$?  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  

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