In[50]:= Assuming[Re@z > Sqrt[1/4 + (Im@z)^2/3], FullSimplify[1/2 ((-3 z + 4 z^3 - Sqrt[-1 + (3 z - 4 z^3)^2])^( 1/3) + (-3 z + 4 z^3 + Sqrt[-1 + (3 z - 4 z^3)^2])^(1/3))]] Out[50]= 1/2 ((-3 z + 4 z^3 - Sqrt[-1 + (3 z - 4 z^3)^2])^( 1/3) + (-3 z + 4 z^3 + Sqrt[-1 + (3 z - 4 z^3)^2])^(1/3)) "Should" be z. Some evidence: In[51]:= FullSimplify[%/.z->1/2] Out[51]= (-1)^(1/3) (*Isolated discontinuity*) In[52]:= FullSimplify[%50/.z->11/21] Out[52]= 11/21 In[53]:= FullSimplify[%50/.z->10001/20001] Out[53]= 10001/20001 In[71]:= Series[%50,{z,1/2,9}]//FullSimplify Out[71]= 1/2+(z-1/2)+O[z-1/2]^10 Not f(1/2) + ... ! In[74]:= Series[%50,{z,1,9}]//FullSimplify Out[74]= 1+(z-1)+O[z-1]^(19/2) In[75]:= Series[%50,{z,3/2,9}]//FullSimplify Out[75]= 3/2+(z-3/2)+O[z-3/2]^10 In[76]:= Series[%50,{z,1+I,9}]//FullSimplify Out[76]= (1+I)+(z-(1+I))+O[z-(1+I)]^10. --Bill Gosper