Trac #20051: Merge #20049 and fix singularity

src/sage/rings/asymptotic/asymptotic_expansion_generators.py

@@ -591,19 +591,19 @@ def SingularityAnalysis(var, zeta=1, alpha=0, beta=0, delta=0,
 
         .. MATH::
 
-            [z^n] \left(\frac{1}{1-z}\right)^\alpha
-            \left(\frac{1}{z} \log \frac{1}{1-z}\right)^\beta
-            \left(\frac{1}{z} \log
-            \left(\frac{1}{z} \log \frac{1}{1-z}\right)\right)^\delta
+            [z^n] \left(\frac{1}{1-z/\zeta}\right)^\alpha
+            \left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)^\beta
+            \left(\frac{1}{z/\zeta} \log
+            \left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)\right)^\delta
 
         (if ``normalized=True``, the default) or
 
         .. MATH::
 
-            [z^n] \left(\frac{1}{1-z}\right)^\alpha
-            \left(\log \frac{1}{1-z}\right)^\beta
+            [z^n] \left(\frac{1}{1-z/\zeta}\right)^\alpha
+            \left(\log \frac{1}{1-z/\zeta}\right)^\beta
             \left(\log
-            \left(\frac{1}{z} \log \frac{1}{1-z}\right)\right)^\delta
+            \left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)\right)^\delta
 
         (if ``normalized=False``).