Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

48 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
The factor 2 is due to the fact that on the Riemann surface (see Figure
1) one also has to integrate the corresponding return path in the lower
sheet for calculating the moduli. With the general result a
further simplification of the formulae is possible.
The imaginary parts of the equations follow directly from (44), (45) and
(46).
The symbol in stands for “+ real infinity”.
Using symmetry properties the functions and (4) can be calculated:
3. TRANSFORMATION OF THE THETA
FORMULA
Now, using relations between Rosenhain’s theta functions, the solution
for the Ernst potential is transformed in such a way that all arguments
become real.