Cauchy's functional equation from complex to reals Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Overview of basic facts about Cauchy functional equationCauchy functional equation over the complex fieldComplex integral revision, this is just Cauchy's Theorem right?How bad can perturbing a functional equation really make things?Solution of Cauchy functional equation which has an antiderivativeOverview of basic facts about Cauchy functional equationComplex number field: ''essentially'' unique?Functional equation $f(x+y)-f(x)-f(y)=alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditionsdiscontinuous solutions of functional equationSolving a multiplicative functional equationAn interesting twist on Cauchy's functional equationCauchy type equation in three variables
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Cauchy's functional equation from complex to reals
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Overview of basic facts about Cauchy functional equationCauchy functional equation over the complex fieldComplex integral revision, this is just Cauchy's Theorem right?How bad can perturbing a functional equation really make things?Solution of Cauchy functional equation which has an antiderivativeOverview of basic facts about Cauchy functional equationComplex number field: ''essentially'' unique?Functional equation $f(x+y)-f(x)-f(y)=alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditionsdiscontinuous solutions of functional equationSolving a multiplicative functional equationAn interesting twist on Cauchy's functional equationCauchy type equation in three variables
$begingroup$
I am looking for the general continuous solutions $f:D rightarrow mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=x$. (Complex to reals would also help).
I tried writing out the functional equation in matrix representation and linking it to solutions in Aczel's textbook. However, the most I got out of this approach was that $f$ is a group homomorphism (well, yes...). The chapter on Cauchy's equation for complex numbers does not seem to address this problem at all. Related questions such as this or this did not clarify my question either.
complex-analysis functional-equations
asked Apr 1 at 14:29
HRSEHRSE
233110
$endgroup$
add a comment |
$begingroup$
I am looking for the general continuous solutions $f:D rightarrow mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=x$. (Complex to reals would also help).
I tried writing out the functional equation in matrix representation and linking it to solutions in Aczel's textbook. However, the most I got out of this approach was that $f$ is a group homomorphism (well, yes...). The chapter on Cauchy's equation for complex numbers does not seem to address this problem at all. Related questions such as this or this did not clarify my question either.
complex-analysis functional-equations
asked Apr 1 at 14:29
HRSEHRSE
233110
$endgroup$
add a comment |
$begingroup$
I am looking for the general continuous solutions $f:D rightarrow mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=x$. (Complex to reals would also help).
I tried writing out the functional equation in matrix representation and linking it to solutions in Aczel's textbook. However, the most I got out of this approach was that $f$ is a group homomorphism (well, yes...). The chapter on Cauchy's equation for complex numbers does not seem to address this problem at all. Related questions such as this or this did not clarify my question either.
complex-analysis functional-equations
asked Apr 1 at 14:29
HRSEHRSE
233110
$endgroup$
I am looking for the general continuous solutions $f:D rightarrow mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=x$. (Complex to reals would also help).
I tried writing out the functional equation in matrix representation and linking it to solutions in Aczel's textbook. However, the most I got out of this approach was that $f$ is a group homomorphism (well, yes...). The chapter on Cauchy's equation for complex numbers does not seem to address this problem at all. Related questions such as this or this did not clarify my question either.
complex-analysis functional-equations
complex-analysis functional-equations
asked Apr 1 at 14:29
HRSEHRSE
233110
asked Apr 1 at 14:29
HRSEHRSE
233110
asked Apr 1 at 14:29
HRSEHRSE
233110
asked Apr 1 at 14:29
HRSEHRSE
233110
asked Apr 1 at 14:29
HRSEHRSE
233110
233110
add a comment |
add a comment |
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