Sun Nov 1 19:27:57 MST 1998 euler% math Mathematica 3.0 for Solaris Copyright 1988-97 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Determining Zero Equivalence ---------- *) 0. Second In[6]:= (* The following expressions are all equal to zero *) 0. Second In[7]:= Sqrt[997] - (997^3)^(1/6) 0. Second Out[7]= 0 In[8]:= Sqrt[999983] - (999983^3)^(1/6) 0.02 Second Out[8]= 0 In[9]:= (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6 0. Second 1/3 2/3 1/3 2/3 3 Out[9]= -6 - 6 (2 + 2 ) + (2 + 2 ) In[10]:= Simplify[%] 0.01 Second Out[10]= 0 In[11]:= Cos[x]^3 + Cos[x]*Sin[x]^2 - Cos[x] 0. Second 3 2 Out[11]= -Cos[x] + Cos[x] + Cos[x] Sin[x] In[12]:= Simplify[%] 0.01 Second Out[12]= 0 In[13]:= (* See Joel Moses, ``Algebraic Simplification: A Guide for the\ > Perplexed'', _Communications of the Association of Computing Machinery_, Volume 14, Number 8, August 1971, 527--537. This expression is zero if Re(x) is contained in the interval ((4 n - 1)/2 pi, (4 n + 1)/2 pi) for n an\ > integer: ..., (-5/2 pi, -3/2 pi), (-pi/2, pi/2), (3/2 pi, 5/2 pi), ... *) 0. Second In[14]:= expr = Log[Tan[1/2*x + Pi/4]] - ArcSinh[Tan[x]] 0. Second Pi x Out[14]= -ArcSinh[Tan[x]] + Log[Tan[-- + -]] 4 2 In[15]:= Simplify[FullSimplify[TrigToExp[expr]]] 6.26 Second 2 I x 2 E 2 I Out[15]= Log[-I + ---------] - Log[-I + 2 Sqrt[-------------] + ----------] I x 2 I x 2 2 I x -I + E (1 + E ) 1 + E In[16]:= (* Use a roundabout method---show that expr is a constant equal to\ > zero *) 0. Second In[17]:= D[expr, x] 0. Second Pi x Pi x Csc[-- + -] Sec[-- + -] 2 4 2 4 2 Sec[x] Out[17]= ----------------------- - ----------------- 2 2 Sqrt[1 + Tan[x] ] In[18]:= Simplify[%] 0.57 Second 2 Out[18]= Sec[x] - Sqrt[Sec[x] ] In[19]:= PowerExpand[%] 0. Second Out[19]= 0 In[20]:= expr /. x -> 0 0. Second Out[20]= 0 In[21]:= Clear[expr] 0. Second In[22]:= Log[(2*Sqrt[r] + 1)/Sqrt[4*r + 4*Sqrt[r] + 1]] 0. Second 1 + 2 Sqrt[r] Out[22]= Log[-------------------------] Sqrt[1 + 4 Sqrt[r] + 4 r] In[23]:= PowerExpand[Simplify[%]] 0.08 Second Out[23]= 0 In[24]:= (4*r + 4*Sqrt[r] + 1)^(Sqrt[r]/(2*Sqrt[r] + 1)) * (2*Sqrt[r] + 1)^(1/(2*Sqrt[r] + 1)) - 2*Sqrt[r] - 1 0. Second 1/(1 + 2 Sqrt[r]) Out[24]= -1 - 2 Sqrt[r] + (1 + 2 Sqrt[r]) Sqrt[r]/(1 + 2 Sqrt[r]) > (1 + 4 Sqrt[r] + 4 r) In[25]:= Simplify[PowerExpand[Simplify[%]]] 0.29 Second Out[25]= 0 In[26]:= (* [Gradshteyn and Ryzhik 9.535(3)] *) 0. Second In[27]:= 2^(1 - z)*Gamma[z]*Zeta[z]*Cos[z*Pi/2] - Pi^z*Zeta[1 - z] 0. Second z 1 - z Pi z Out[27]= -(Pi Zeta[1 - z]) + 2 Cos[----] Gamma[z] Zeta[z] 2 In[28]:= FullSimplify[%] 1.15 Second Out[28]= 0 In[29]:= (* ---------- Quit ---------- *) 0. Second In[30]:= Quit[] real 11.19 user 9.28 sys 0.27