quasiperiod and half quasiperiod relations for Jacobi ϑ functions

quasiperiod and half quasiperiod relations for Jacobi

ϑ

functions

The

theta functions

have

quasiperiods

π

and

π

⁢

τ

:

θ

1

(

z

∣

τ

)

=

-

θ

1

(

z

+

π

∣

τ

)

=

-

e

2

⁢

i

⁢

z

+

i

⁢

π

⁢

τ

θ

1

(

z

+

π

τ

∣

τ

)

θ

2

(

z

∣

τ

)

=

-

θ

2

(

z

+

π

∣

τ

)

=

e

2

⁢

i

⁢

z

+

i

⁢

π

⁢

τ

θ

2

(

z

+

π

τ

∣

τ

)

θ

3

(

z

∣

τ

)

=

θ

3

(

z

+

π

∣

τ

)

=

e

2

⁢

i

⁢

z

+

i

⁢

π

⁢

τ

θ

3

(

z

+

π

τ

∣

τ

)

θ

4

(

z

∣

τ

)

=

θ

4

(

z

+

π

∣

τ

)

=

-

e

2

⁢

i

⁢

z

+

i

⁢

π

⁢

τ

θ

4

(

z

+

π

τ

∣

τ

)

By adding half a quasiperiod, one can can convert one theta function into another.

θ

1

(

z

∣

τ

)

=

-

θ

2

(

z

+

π

/

2

∣

τ

)

=

-

i

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

4

(

z

+

π

τ

/

2

∣

τ

)

=

-

i

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

3

(

z

+

π

/

2

+

π

τ

/

2

∣

τ

)

θ

2

(

z

∣

τ

)

=

θ

1

(

z

+

π

/

2

∣

τ

)

=

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

3

(

z

+

π

τ

/

2

∣

τ

)

=

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

4

(

z

+

π

/

2

+

π

τ

/

2

∣

τ

)

θ

3

(

z

∣

τ

)

=

θ

4

(

z

+

π

/

2

∣

τ

)

=

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

2

(

z

+

π

τ

/

2

∣

τ

)

=

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

1

(

z

+

π

/

2

+

π

τ

/

2

∣

τ

)

θ

4

(

z

∣

τ

)

=

θ

3

(

z

+

π

/

2

∣

τ

)

=

-

i

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

1

(

z

+

π

τ

/

2

∣

τ

)

=

i

e

i

⁢

z

+

i

⁢

π

⁢

τ

/

4

θ

2

(

z

+

π

/

2

+

π

τ

/

2

∣

τ

)

The proof of these

identities

is simply a matter of adding the appropriate quasiperiod to

z

in the series defining the theta function and using the

addition formula

for the

exponential

to simplify the result. For instance,

θ

2

(

z

+

π

τ

/

2

∣

τ

)

=

∑

n

=

-

∞

+

∞

e

i

⁢

π

⁢

τ

⁢

(

n

+

1

/

2

)

2

+

(

2

⁢

n

+

1

)

⁢

i

⁢

(

z

+

π

⁢

τ

/

2

)

Multiplying out,

π

⁢

τ

⁢

(

n

+

1

/

2

)

2

+

(

2

⁢

n

+

1

)

⁢

(

z

+

π

⁢

τ

/

2

)

=

π

⁢

τ

⁢

n

2

+

2

⁢

n

⁢

z

+

2

⁢

π

⁢

τ

⁢

n

+

z

+

3

⁢

π

⁢

τ

/

4

=

π

⁢

τ

⁢

(

n

+

1

)

2

+

2

⁢

(

n

+

1

)

⁢

z

-

z

-

π

⁢

τ

/

4

and hence,

∑

n

=

-

∞

+

∞

e

i

⁢

π

⁢

τ

⁢

(

n

+

1

/

2

)

2

+

(

2

⁢

n

+

1

)

⁢

i

⁢

(

z

+

π

⁢

τ

/

2

)

=

e

-

i

⁢

z

-

i

⁢

π

⁢

τ

/

4

⁢

∑

n

=

-

∞

+

∞

e

i

⁢

π

⁢

τ

⁢

(

n

+

1

)

2

+

2

⁢

i

⁢

(

n

+

1

)

⁢

z

=

e

-

i

⁢

z

-

i

⁢

π

⁢

τ

/

4

∑

n

=

-

∞

+

∞

e

i

⁢

π

⁢

τ

⁢

n

2

+

2

⁢

i

⁢

n

⁢

z

=

e

-

i

⁢

z

-

i

⁢

π

⁢

τ

/

4

θ

3

(

z

∣

τ

)

Once enough half quasiperiod relations have been verified in this manner, the remaining relations may be deduced from them. Finally, the quasiperiod relations may be deduced from the half quasiperiod relations.

An important use of these relations is in constructing

elliptic functions

. By taking suitable quotients of theta functions with the same quasiperiod, one can arrange for the result to be periodic, not just quasi-periodic.

Title

quasiperiod and half quasiperiod relations for Jacobi

ϑ

functions

Canonical name

QuasiperiodAndHalfQuasiperiodRelationsForJacobivarthetaFunctions

Date of creation

2013-03-22 14:40:27

Last modified on

2013-03-22 14:40:27

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 35H30

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