Elliptic function with essential singularity
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The Jacobi theta function, $theta(u;tau)$ (in some convention which will be implicit below), has the following elliptic transformation behavior:
$$ theta(u+ m + n tau;tau) = (-1)^{m+n} e^{2 pi i (-n u - frac{1}{2} n^2 tau)} theta(u,tau) $$
In particular, this means that $log theta(u;tau)$ picks up a linear shift under such a transformation. Then if we take two derivatives with respect to $u$, $(log theta(u;tau))''$, this kills this linear piece and we find an elliptic function with a double pole at $u=0$, which is precisely the Weierstrass p-function. On the other hand, if we take a single derivative, $(log theta(u;tau))' = theta'(u;tau)/theta(u;tau)$, we get a function with a single simple pole at $u=0$, but this can still pick up shifts by integer multiples of $2 pi i$ under elliptic transformations. Then it is natural to consider the exponential:
$$ f(u) = exp bigg(frac{theta'(u;tau)}{theta(u;tau)} bigg) $$
Then $f(u)$ is an elliptic function with an essential singularity at $u=0$. I was wondering if this function (or its log) has any special name or significance in the study of elliptic functions.
complex-analysis
asked Dec 4 at 20:15
user6013
1734
add a comment |
up vote
3
down vote
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The Jacobi theta function, $theta(u;tau)$ (in some convention which will be implicit below), has the following elliptic transformation behavior:
$$ theta(u+ m + n tau;tau) = (-1)^{m+n} e^{2 pi i (-n u - frac{1}{2} n^2 tau)} theta(u,tau) $$
In particular, this means that $log theta(u;tau)$ picks up a linear shift under such a transformation. Then if we take two derivatives with respect to $u$, $(log theta(u;tau))''$, this kills this linear piece and we find an elliptic function with a double pole at $u=0$, which is precisely the Weierstrass p-function. On the other hand, if we take a single derivative, $(log theta(u;tau))' = theta'(u;tau)/theta(u;tau)$, we get a function with a single simple pole at $u=0$, but this can still pick up shifts by integer multiples of $2 pi i$ under elliptic transformations. Then it is natural to consider the exponential:
$$ f(u) = exp bigg(frac{theta'(u;tau)}{theta(u;tau)} bigg) $$
Then $f(u)$ is an elliptic function with an essential singularity at $u=0$. I was wondering if this function (or its log) has any special name or significance in the study of elliptic functions.
complex-analysis
asked Dec 4 at 20:15
user6013
1734
the log of $f$ is, up to a linear function, the Weiestrass zeta function
– user8268
Dec 5 at 18:02
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
The Jacobi theta function, $theta(u;tau)$ (in some convention which will be implicit below), has the following elliptic transformation behavior:
$$ theta(u+ m + n tau;tau) = (-1)^{m+n} e^{2 pi i (-n u - frac{1}{2} n^2 tau)} theta(u,tau) $$
In particular, this means that $log theta(u;tau)$ picks up a linear shift under such a transformation. Then if we take two derivatives with respect to $u$, $(log theta(u;tau))''$, this kills this linear piece and we find an elliptic function with a double pole at $u=0$, which is precisely the Weierstrass p-function. On the other hand, if we take a single derivative, $(log theta(u;tau))' = theta'(u;tau)/theta(u;tau)$, we get a function with a single simple pole at $u=0$, but this can still pick up shifts by integer multiples of $2 pi i$ under elliptic transformations. Then it is natural to consider the exponential:
$$ f(u) = exp bigg(frac{theta'(u;tau)}{theta(u;tau)} bigg) $$
Then $f(u)$ is an elliptic function with an essential singularity at $u=0$. I was wondering if this function (or its log) has any special name or significance in the study of elliptic functions.
complex-analysis
asked Dec 4 at 20:15
user6013
1734
The Jacobi theta function, $theta(u;tau)$ (in some convention which will be implicit below), has the following elliptic transformation behavior:
$$ theta(u+ m + n tau;tau) = (-1)^{m+n} e^{2 pi i (-n u - frac{1}{2} n^2 tau)} theta(u,tau) $$
In particular, this means that $log theta(u;tau)$ picks up a linear shift under such a transformation. Then if we take two derivatives with respect to $u$, $(log theta(u;tau))''$, this kills this linear piece and we find an elliptic function with a double pole at $u=0$, which is precisely the Weierstrass p-function. On the other hand, if we take a single derivative, $(log theta(u;tau))' = theta'(u;tau)/theta(u;tau)$, we get a function with a single simple pole at $u=0$, but this can still pick up shifts by integer multiples of $2 pi i$ under elliptic transformations. Then it is natural to consider the exponential:
$$ f(u) = exp bigg(frac{theta'(u;tau)}{theta(u;tau)} bigg) $$
Then $f(u)$ is an elliptic function with an essential singularity at $u=0$. I was wondering if this function (or its log) has any special name or significance in the study of elliptic functions.
complex-analysis
complex-analysis
asked Dec 4 at 20:15
user6013
1734
asked Dec 4 at 20:15
user6013
1734
asked Dec 4 at 20:15
user6013
1734
asked Dec 4 at 20:15
user6013
1734
asked Dec 4 at 20:15
user6013
1734
1734
the log of $f$ is, up to a linear function, the Weiestrass zeta function
– user8268
Dec 5 at 18:02
add a comment |
the log of $f$ is, up to a linear function, the Weiestrass zeta function
– user8268
Dec 5 at 18:02
the log of $f$ is, up to a linear function, the Weiestrass zeta function
– user8268
Dec 5 at 18:02
the log of $f$ is, up to a linear function, the Weiestrass zeta function
– user8268
Dec 5 at 18:02
add a comment |
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the log of $f$ is, up to a linear function, the Weiestrass zeta function
– user8268
Dec 5 at 18:02