Mixing
can we define logarithm function $log$ on $mathbb C((T))$
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let $mathbb C[[T]]$ be the ring of formal power series in one formal variable $T$, and $mathbb C((T))$ be its fraction field. At first we can definitely define the exponential function $exp:mathbb C((T))to mathbb C((T))$ by
$$
fmapsto exp(f):= sum_k=0^inftyfracf^kk!
$$
However it seems there are troubles in defining $log$. For example, for any $a$ we can naively 'put' $log(f)=log(a)+log(1+fracf-aa)$ and in a similar way we may use the Taylor expansion to the last term.
Note that a meaningful definition of $log$ should satisfy $log circ
exp =id$ or $exp circ log=id$.
So, it seems that the definition depends on $a$, right or not? Let me put the question simpler: is it possible to define $log(T)$ in $mathbb C((T))$?
logarithms power-series exponential-function
asked Jul 18 at 17:55
Hang
395214
add a comment |Â
up vote
0
down vote
favorite
Let $mathbb C[[T]]$ be the ring of formal power series in one formal variable $T$, and $mathbb C((T))$ be its fraction field. At first we can definitely define the exponential function $exp:mathbb C((T))to mathbb C((T))$ by
$$
fmapsto exp(f):= sum_k=0^inftyfracf^kk!
$$
However it seems there are troubles in defining $log$. For example, for any $a$ we can naively 'put' $log(f)=log(a)+log(1+fracf-aa)$ and in a similar way we may use the Taylor expansion to the last term.
Note that a meaningful definition of $log$ should satisfy $log circ
exp =id$ or $exp circ log=id$.
So, it seems that the definition depends on $a$, right or not? Let me put the question simpler: is it possible to define $log(T)$ in $mathbb C((T))$?
logarithms power-series exponential-function
asked Jul 18 at 17:55
Hang
395214
Possibly relevant: math.stackexchange.com/questions/2486706/….
– MartÃn-Blas Pérez Pinilla
Jul 21 at 16:49
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $mathbb C[[T]]$ be the ring of formal power series in one formal variable $T$, and $mathbb C((T))$ be its fraction field. At first we can definitely define the exponential function $exp:mathbb C((T))to mathbb C((T))$ by
$$
fmapsto exp(f):= sum_k=0^inftyfracf^kk!
$$
However it seems there are troubles in defining $log$. For example, for any $a$ we can naively 'put' $log(f)=log(a)+log(1+fracf-aa)$ and in a similar way we may use the Taylor expansion to the last term.
Note that a meaningful definition of $log$ should satisfy $log circ
exp =id$ or $exp circ log=id$.
So, it seems that the definition depends on $a$, right or not? Let me put the question simpler: is it possible to define $log(T)$ in $mathbb C((T))$?
logarithms power-series exponential-function
asked Jul 18 at 17:55
Hang
395214
Let $mathbb C[[T]]$ be the ring of formal power series in one formal variable $T$, and $mathbb C((T))$ be its fraction field. At first we can definitely define the exponential function $exp:mathbb C((T))to mathbb C((T))$ by
$$
fmapsto exp(f):= sum_k=0^inftyfracf^kk!
$$
However it seems there are troubles in defining $log$. For example, for any $a$ we can naively 'put' $log(f)=log(a)+log(1+fracf-aa)$ and in a similar way we may use the Taylor expansion to the last term.
Note that a meaningful definition of $log$ should satisfy $log circ
exp =id$ or $exp circ log=id$.
So, it seems that the definition depends on $a$, right or not? Let me put the question simpler: is it possible to define $log(T)$ in $mathbb C((T))$?
logarithms power-series exponential-function
asked Jul 18 at 17:55
Hang
395214
asked Jul 18 at 17:55
Hang
395214
asked Jul 18 at 17:55
Hang
395214
asked Jul 18 at 17:55
Hang
395214
395214
Possibly relevant: math.stackexchange.com/questions/2486706/….
– MartÃn-Blas Pérez Pinilla
Jul 21 at 16:49
add a comment |Â
Possibly relevant: math.stackexchange.com/questions/2486706/….
– MartÃn-Blas Pérez Pinilla
Jul 21 at 16:49
Possibly relevant: math.stackexchange.com/questions/2486706/….
– MartÃn-Blas Pérez Pinilla
Jul 21 at 16:49
Possibly relevant: math.stackexchange.com/questions/2486706/….
– MartÃn-Blas Pérez Pinilla
Jul 21 at 16:49
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
Too long for a comment:
At first we can definitely define the exponential function...
In fact, you can't define the formal exponential even in $Bbb C[[T]]$. If $f(T) = a_0 + a_1T + cdots$ with $a_0ne 0$, the 0-th term of $exp(f)$ is the infinite sum
$$sum_k=0^inftyfraca_0^kk!.$$
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
But we are working with $mathbb C$, and this is just $e^a_0in mathbb C$, right?
– Hang
Jul 23 at 17:44
@Hang, formal series means always finite sums.
– MartÃn-Blas Pérez Pinilla
Jul 24 at 8:34
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Too long for a comment:
At first we can definitely define the exponential function...
In fact, you can't define the formal exponential even in $Bbb C[[T]]$. If $f(T) = a_0 + a_1T + cdots$ with $a_0ne 0$, the 0-th term of $exp(f)$ is the infinite sum
$$sum_k=0^inftyfraca_0^kk!.$$
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
But we are working with $mathbb C$, and this is just $e^a_0in mathbb C$, right?
– Hang
Jul 23 at 17:44
@Hang, formal series means always finite sums.
– MartÃn-Blas Pérez Pinilla
Jul 24 at 8:34
add a comment |Â
up vote
0
down vote
Too long for a comment:
At first we can definitely define the exponential function...
In fact, you can't define the formal exponential even in $Bbb C[[T]]$. If $f(T) = a_0 + a_1T + cdots$ with $a_0ne 0$, the 0-th term of $exp(f)$ is the infinite sum
$$sum_k=0^inftyfraca_0^kk!.$$
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
But we are working with $mathbb C$, and this is just $e^a_0in mathbb C$, right?
– Hang
Jul 23 at 17:44
@Hang, formal series means always finite sums.
– MartÃn-Blas Pérez Pinilla
Jul 24 at 8:34
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Too long for a comment:
At first we can definitely define the exponential function...
In fact, you can't define the formal exponential even in $Bbb C[[T]]$. If $f(T) = a_0 + a_1T + cdots$ with $a_0ne 0$, the 0-th term of $exp(f)$ is the infinite sum
$$sum_k=0^inftyfraca_0^kk!.$$
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
Too long for a comment:
At first we can definitely define the exponential function...
In fact, you can't define the formal exponential even in $Bbb C[[T]]$. If $f(T) = a_0 + a_1T + cdots$ with $a_0ne 0$, the 0-th term of $exp(f)$ is the infinite sum
$$sum_k=0^inftyfraca_0^kk!.$$
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
answered Jul 21 at 17:00
MartÃn-Blas Pérez Pinilla
33.4k42570
33.4k42570
But we are working with $mathbb C$, and this is just $e^a_0in mathbb C$, right?
– Hang
Jul 23 at 17:44
@Hang, formal series means always finite sums.
– MartÃn-Blas Pérez Pinilla
Jul 24 at 8:34
add a comment |Â
But we are working with $mathbb C$, and this is just $e^a_0in mathbb C$, right?
– Hang
Jul 23 at 17:44
@Hang, formal series means always finite sums.
– MartÃn-Blas Pérez Pinilla
Jul 24 at 8:34
But we are working with $mathbb C$, and this is just $e^a_0in mathbb C$, right?
– Hang
Jul 23 at 17:44
But we are working with $mathbb C$, and this is just $e^a_0in mathbb C$, right?
– Hang
Jul 23 at 17:44
@Hang, formal series means always finite sums.
– MartÃn-Blas Pérez Pinilla
Jul 24 at 8:34
@Hang, formal series means always finite sums.
– MartÃn-Blas Pérez Pinilla
Jul 24 at 8:34
add a comment |Â
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%2f2855836%2fcan-we-define-logarithm-function-log-on-mathbb-ct%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
Possibly relevant: math.stackexchange.com/questions/2486706/….
– MartÃn-Blas Pérez Pinilla
Jul 21 at 16:49