add eta() implementing the Dedekind eta function

.gitignore

@@ -50,6 +50,7 @@ doc/source/_build/
 
 # Files generated by setupegg.py
 mpmath.egg-info/
+.eggs
 
 # Coverage-related files
 .coverage

doc/source/functions/elliptic.txt

@@ -91,6 +91,11 @@ Jacobi elliptic functions
 Modular functions
 ......................
 
+:func:`eta`
+^^^^^^^^^^^^^^^
+.. autofunction:: mpmath.eta(tau)
+
 :func:`kleinj`
 ^^^^^^^^^^^^^^^
 .. autofunction:: mpmath.kleinj(tau=None, **kwargs)
+

mpmath/__init__.py

@@ -47,6 +47,7 @@ qbarfrom = mp.qbarfrom
 ellipfun = mp.ellipfun
 jtheta = mp.jtheta
 kleinj = mp.kleinj
+eta = mp.eta
 
 qp = mp.qp
 qhyper = mp.qhyper

mpmath/functions/elliptic.py

@@ -64,6 +64,33 @@ between the various parameters (:func:`~mpmath.qfrom`, :func:`~mpmath.mfrom`,
 
 from .functions import defun, defun_wrapped
 
+@defun_wrapped
+def eta(ctx, tau):
+    r"""
+    Returns the Dedekind eta function of tau in the upper half-plane.
+
+        >>> from mpmath import *
+        >>> mp.dps = 25; mp.pretty = True
+        >>> eta(1j); gamma(0.25) / (2*pi**0.75)
+        (0.7682254223260566590025942 + 0.0j)
+        0.7682254223260566590025942
+        >>> tau = sqrt(2) + sqrt(5)*1j
+        >>> eta(-1/tau); sqrt(-1j*tau) * eta(tau)
+        (0.9022859908439376463573294 + 0.07985093673948098408048575j)
+        (0.9022859908439376463573295 + 0.07985093673948098408048575j)
+        >>> eta(tau+1); exp(pi*1j/12) * eta(tau)
+        (0.4493066139717553786223114 + 0.3290014793877986663915939j)
+        (0.4493066139717553786223114 + 0.3290014793877986663915939j)
+        >>> f = lambda z: diff(eta, z) / eta(z)
+        >>> chop(36*diff(f,tau)**2 - 24*diff(f,tau,2)*f(tau) + diff(f,tau,3))
+        0.0
+
+    """
+    if ctx.im(tau) <= 0.0:
+        raise ValueError("eta is only defined in the upper half-plane")
+    q = ctx.expjpi(tau/12)
+    return q * ctx.qp(q**24)
+
 def nome(ctx, m):
     m = ctx.convert(m)
     if not m: