Let $g$ be real valued and continuous on [0,1]. Show that if $f(z) = int_0^1 g(t) (cos zt) dt $ is 0 for all...
Let $g$ be real valued and continuous on [0,1]. Show that if the complex valued function $$f(z) := int_0^1 g(t) (cos zt) dt $$ is 0 for all $z in mathbb{C}$, then $g$ is identically 0.
I managed to show that $f$ is an entire function by showing its continuity and then using Morera’s Theorem, but know not how to proceed with the assumption that $f$ is 0 on $mathbb{C}$.
I thought that assuming $g$ were not 0 would allow me to locate a $z$ such that $f(z)neq 0$ but I had no such luck. I would appreciate suggestions on what to do next.
complex-analysis analysis
asked Dec 8 at 18:58
F.Tomas
63
add a comment |
Let $g$ be real valued and continuous on [0,1]. Show that if the complex valued function $$f(z) := int_0^1 g(t) (cos zt) dt $$ is 0 for all $z in mathbb{C}$, then $g$ is identically 0.
I managed to show that $f$ is an entire function by showing its continuity and then using Morera’s Theorem, but know not how to proceed with the assumption that $f$ is 0 on $mathbb{C}$.
I thought that assuming $g$ were not 0 would allow me to locate a $z$ such that $f(z)neq 0$ but I had no such luck. I would appreciate suggestions on what to do next.
complex-analysis analysis
asked Dec 8 at 18:58
F.Tomas
63
Is $f$ bounded?
– Randall
Dec 8 at 19:05
Not necessarily, without the assumption that $f$ is identically 0.
– F.Tomas
Dec 8 at 19:37
add a comment |
Let $g$ be real valued and continuous on [0,1]. Show that if the complex valued function $$f(z) := int_0^1 g(t) (cos zt) dt $$ is 0 for all $z in mathbb{C}$, then $g$ is identically 0.
I managed to show that $f$ is an entire function by showing its continuity and then using Morera’s Theorem, but know not how to proceed with the assumption that $f$ is 0 on $mathbb{C}$.
I thought that assuming $g$ were not 0 would allow me to locate a $z$ such that $f(z)neq 0$ but I had no such luck. I would appreciate suggestions on what to do next.
complex-analysis analysis
asked Dec 8 at 18:58
F.Tomas
63
Let $g$ be real valued and continuous on [0,1]. Show that if the complex valued function $$f(z) := int_0^1 g(t) (cos zt) dt $$ is 0 for all $z in mathbb{C}$, then $g$ is identically 0.
I managed to show that $f$ is an entire function by showing its continuity and then using Morera’s Theorem, but know not how to proceed with the assumption that $f$ is 0 on $mathbb{C}$.
I thought that assuming $g$ were not 0 would allow me to locate a $z$ such that $f(z)neq 0$ but I had no such luck. I would appreciate suggestions on what to do next.
complex-analysis analysis
complex-analysis analysis
asked Dec 8 at 18:58
F.Tomas
63
asked Dec 8 at 18:58
F.Tomas
63
edited Dec 8 at 19:30
asked Dec 8 at 18:58
F.Tomas
63
asked Dec 8 at 18:58
F.Tomas
63
asked Dec 8 at 18:58
F.Tomas
63
63
Is $f$ bounded?
– Randall
Dec 8 at 19:05
Not necessarily, without the assumption that $f$ is identically 0.
– F.Tomas
Dec 8 at 19:37
add a comment |
Is $f$ bounded?
– Randall
Dec 8 at 19:05
Not necessarily, without the assumption that $f$ is identically 0.
– F.Tomas
Dec 8 at 19:37
Is $f$ bounded?
– Randall
Dec 8 at 19:05
Is $f$ bounded?
– Randall
Dec 8 at 19:05
Not necessarily, without the assumption that $f$ is identically 0.
– F.Tomas
Dec 8 at 19:37
Not necessarily, without the assumption that $f$ is identically 0.
– F.Tomas
Dec 8 at 19:37
add a comment |
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Is $f$ bounded?
– Randall
Dec 8 at 19:05
Not necessarily, without the assumption that $f$ is identically 0.
– F.Tomas
Dec 8 at 19:37