# Question:wrong series expansion for JacobiTheta3 and JacobiTheta4

## Question:wrong series expansion for JacobiTheta3 and JacobiTheta4

Maple 2020

There is an error in the series expansion for the JacobiTheta3 and JacobiTheta4 function

> q := exp(I*((2*I)*Pi*1.20388508772313))

>series(JacobiTheta3(z, q)-JacobiTheta3(0, q), z, 2);

gives -1+O(z^2)

>series(JacobiTheta3(z, q)-JacobiTheta3(0, q), z, 2);

gives -1+O(z^2)

The issue is that

>series(JacobiTheta3(z, q),z,2);

gives

series(Sum(2*Q^(_k1^2),_k1 = 1 .. infinity)+O(z^2),z,2)

where the term n=0 is missing.

The same for JacobiTheta4

series(JacobiTheta4(z, Q), z, 2);

series(Sum(2*(-1)^_k1*Q^(_k1^2),_k1 = 1 .. infinity)+O(z^2),z,2)

The same issue arises when taking the limit

limit(JacobiTheta3(z, sqrt(q))-JacobiTheta3(0, sqrt(q)), z=0);

gives -1

same for  limit(JacobiTheta4(z, sqrt(q))-JacobiTheta4(0, sqrt(q)), z=0); gives -1

This error occurs in maple 2018 and maple 2020.

On the other side when taking numering values approching 0 there is no problems.

So it seems that series and limit uses some codes wrong formulae.

How can one patch this?