Showing that for any $epsilon > 0$, the function $frac1z + i + sin(z)$ has an infinite number of zeros.
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Problem:
Prove that, for any given $epsilon > 0$, the function $$ z mapsto frac1z + i + sin(z) $$ has an infinite number of zeros in the strip $|textIm(z) | < epsilon$.
Attempt: I was trying to apply Rouche's theorem to this problem. But to apply this theorem, I need to specify an open set containing a circle $C$ and its interior. Then I can compare the number of zeros of holomorphic functions $f$ and $f + g$ inside this circle. However, I'm confused because here the region is an infinite strip.
In this case, I would let $f(z) = sin(z)$ and $g(z) = frac1z+i$. I was looking to bound $g(z)$ by $f(z)$ somehow and then argue that since $sin(z)$ has an infinite number of zeros, so does the stated function.
Any help is appreciated.
complex-analysis
edited Aug 17 at 4:06
Nosrati
20.7k41644
asked Aug 15 at 9:26
Kamil
1,90421237
add a comment |Â
up vote
1
down vote
favorite
Problem:
Prove that, for any given $epsilon > 0$, the function $$ z mapsto frac1z + i + sin(z) $$ has an infinite number of zeros in the strip $|textIm(z) | < epsilon$.
Attempt: I was trying to apply Rouche's theorem to this problem. But to apply this theorem, I need to specify an open set containing a circle $C$ and its interior. Then I can compare the number of zeros of holomorphic functions $f$ and $f + g$ inside this circle. However, I'm confused because here the region is an infinite strip.
In this case, I would let $f(z) = sin(z)$ and $g(z) = frac1z+i$. I was looking to bound $g(z)$ by $f(z)$ somehow and then argue that since $sin(z)$ has an infinite number of zeros, so does the stated function.
Any help is appreciated.
complex-analysis
edited Aug 17 at 4:06
Nosrati
20.7k41644
asked Aug 15 at 9:26
Kamil
1,90421237
1
Look at $f$ and $g$ on circles of radius $epsilon$ around the zeroes of $f$. You should aim to apply Rouché's theorem on all but finitely many of them.
– mercio
Aug 15 at 10:03
3
Possible duplicate of Prove that $frac1z+i +sin(z)=0$ has infinite solutions over $mathbbC$
– Nosrati
Aug 17 at 4:03
Also see math.stackexchange.com/questions/2759800/…
– Nosrati
Aug 17 at 4:04
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Problem:
Prove that, for any given $epsilon > 0$, the function $$ z mapsto frac1z + i + sin(z) $$ has an infinite number of zeros in the strip $|textIm(z) | < epsilon$.
Attempt: I was trying to apply Rouche's theorem to this problem. But to apply this theorem, I need to specify an open set containing a circle $C$ and its interior. Then I can compare the number of zeros of holomorphic functions $f$ and $f + g$ inside this circle. However, I'm confused because here the region is an infinite strip.
In this case, I would let $f(z) = sin(z)$ and $g(z) = frac1z+i$. I was looking to bound $g(z)$ by $f(z)$ somehow and then argue that since $sin(z)$ has an infinite number of zeros, so does the stated function.
Any help is appreciated.
complex-analysis
edited Aug 17 at 4:06
Nosrati
20.7k41644
asked Aug 15 at 9:26
Kamil
1,90421237
Problem:
Prove that, for any given $epsilon > 0$, the function $$ z mapsto frac1z + i + sin(z) $$ has an infinite number of zeros in the strip $|textIm(z) | < epsilon$.
Attempt: I was trying to apply Rouche's theorem to this problem. But to apply this theorem, I need to specify an open set containing a circle $C$ and its interior. Then I can compare the number of zeros of holomorphic functions $f$ and $f + g$ inside this circle. However, I'm confused because here the region is an infinite strip.
In this case, I would let $f(z) = sin(z)$ and $g(z) = frac1z+i$. I was looking to bound $g(z)$ by $f(z)$ somehow and then argue that since $sin(z)$ has an infinite number of zeros, so does the stated function.
Any help is appreciated.
complex-analysis
edited Aug 17 at 4:06
Nosrati
20.7k41644
asked Aug 15 at 9:26
Kamil
1,90421237
edited Aug 17 at 4:06
Nosrati
20.7k41644
edited Aug 17 at 4:06
Nosrati
20.7k41644
edited Aug 17 at 4:06
Nosrati
20.7k41644
20.7k41644
asked Aug 15 at 9:26
Kamil
1,90421237
asked Aug 15 at 9:26
Kamil
1,90421237
asked Aug 15 at 9:26
Kamil
1,90421237
1,90421237
1
Look at $f$ and $g$ on circles of radius $epsilon$ around the zeroes of $f$. You should aim to apply Rouché's theorem on all but finitely many of them.
– mercio
Aug 15 at 10:03
3
Possible duplicate of Prove that $frac1z+i +sin(z)=0$ has infinite solutions over $mathbbC$
– Nosrati
Aug 17 at 4:03
Also see math.stackexchange.com/questions/2759800/…
– Nosrati
Aug 17 at 4:04
add a comment |Â
1
Look at $f$ and $g$ on circles of radius $epsilon$ around the zeroes of $f$. You should aim to apply Rouché's theorem on all but finitely many of them.
– mercio
Aug 15 at 10:03
3
Possible duplicate of Prove that $frac1z+i +sin(z)=0$ has infinite solutions over $mathbbC$
– Nosrati
Aug 17 at 4:03
Also see math.stackexchange.com/questions/2759800/…
– Nosrati
Aug 17 at 4:04
1
1
Look at $f$ and $g$ on circles of radius $epsilon$ around the zeroes of $f$. You should aim to apply Rouché's theorem on all but finitely many of them.
– mercio
Aug 15 at 10:03
Look at $f$ and $g$ on circles of radius $epsilon$ around the zeroes of $f$. You should aim to apply Rouché's theorem on all but finitely many of them.
– mercio
Aug 15 at 10:03
3
3
Possible duplicate of Prove that $frac1z+i +sin(z)=0$ has infinite solutions over $mathbbC$
– Nosrati
Aug 17 at 4:03
Possible duplicate of Prove that $frac1z+i +sin(z)=0$ has infinite solutions over $mathbbC$
– Nosrati
Aug 17 at 4:03
Also see math.stackexchange.com/questions/2759800/…
– Nosrati
Aug 17 at 4:04
Also see math.stackexchange.com/questions/2759800/…
– Nosrati
Aug 17 at 4:04
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2883392%2fshowing-that-for-any-epsilon-0-the-function-frac1z-i-sinz-ha%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Look at $f$ and $g$ on circles of radius $epsilon$ around the zeroes of $f$. You should aim to apply Rouché's theorem on all but finitely many of them.
– mercio
Aug 15 at 10:03
3
Possible duplicate of Prove that $frac1z+i +sin(z)=0$ has infinite solutions over $mathbbC$
– Nosrati
Aug 17 at 4:03
Also see math.stackexchange.com/questions/2759800/…
– Nosrati
Aug 17 at 4:04