- #1
mnb96
- 715
- 5
Hello,
I want to find a family of functions [itex]\phi:\mathbb{R} \rightarrow \mathbb{C}[/itex] that have the property: [tex]\phi(x+y)=\phi(x)\phi(y)[/tex] where [itex]x,y\in \mathbb{R}[/itex].
I know that any exponential function of the kind [itex]\phi(x)=a^x[/itex] with [itex]a\in\mathbb{C}[/itex] will satisfy this property.
Is this the only choice, or are there other functions that I am missing that satisfy the above property?
I want to find a family of functions [itex]\phi:\mathbb{R} \rightarrow \mathbb{C}[/itex] that have the property: [tex]\phi(x+y)=\phi(x)\phi(y)[/tex] where [itex]x,y\in \mathbb{R}[/itex].
I know that any exponential function of the kind [itex]\phi(x)=a^x[/itex] with [itex]a\in\mathbb{C}[/itex] will satisfy this property.
Is this the only choice, or are there other functions that I am missing that satisfy the above property?