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MathGroup Archive 2006

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Re: Troubles with Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70260] Re: Troubles with Integrate
  • From: "astanoff" <astanoff at gmail.com>
  • Date: Tue, 10 Oct 2006 06:12:14 -0400 (EDT)
  • References: <eg84hk$nir$1@smc.vnet.net>

Hi,
The exact value is 4/3 - (8*Sqrt[2])/(9*Pi),
which you can get this way (with appropriate translations) :

In[1]:=psi:= (-Pi^(-1))*((Sin[Pi*((t - 5/4)/3)]/(t - 5/4))*Cos[Pi*(t -
5/4)] +
(Sin[Pi*(t + 1/4)]/(t + 1/4))*Cos[Pi*(t + 1/4)]) +
(1/Pi)*((Sin[2*Pi*((t - 7/8)/3)]/(t - 7/8))*Sin[2*Pi*(t - 7/8)] -
(Sin[2*Pi*((t - 1/8)/3)]/(t - 1/8))*Sin[2*Pi*(t - 1/8)]);
In[2]:=ex=List@@(psi^2//Expand)
...
look at each term to translate accordingly

In[3]:=i[1]=Integrate[ex[[1]]/.t\[Rule]t+5/4,{t,-\[Infinity],\[Infinity]}]
Out[3]=\!\(1\/6\)

In[4]:=i[2]=Integrate[ex[[2]]/.t\[Rule]t+5/4,{t,-\[Infinity],\[Infinity]}]
Out[4]=0

In[5]:=i[3]=Integrate[ex[[3]]/.t\[Rule]t+7/8,{t,-\[Infinity],\[Infinity]}]
Out[5]=\!\(1\/3\)

In[6]:=i[4]=Integrate[ex[[4]]/.t\[Rule]t+5/4,{t,-\[Infinity],\[Infinity]}]
Out[6]=0

In[7]:=i[5]=Integrate[ex[[5]]/.t\[Rule]t+7/8,{t,-\[Infinity],\[Infinity]}]
Out[7]=0

In[8]:=i[6]=Integrate[ex[[6]]/.t\[Rule]t+1/8,{t,-\[Infinity],\[Infinity]}]
Out[8]=\!\(1\/3\)

In[9]:=i[7]=Integrate[ex[[7]]/.t\[Rule]t+5/4,{t,-\[Infinity],\[Infinity]}]
Out[9]=0

In[10]:=i[8]=Integrate[ex[[8]]/.t\[Rule]t+7/8,{t,-\[Infinity],\[Infinity]}]
Out[10]=-((2*Sqrt[2])/(9*Pi))

In[11]:=i[9]=Integrate[ex[[9]]/.t\[Rule]t+1/8,{t,-\[Infinity],\[Infinity]}]
Out[11]=-((2*Sqrt[2])/(3*Pi))

In[12]:=i[10]=Integrate[ex[[10]]/.t\[Rule]t-1/4,{t,-\[Infinity],\[Infinity]}]
Out[12]=\!\(1\/2\)

In[13]:=Sum[i[k],{k,1,10}]
Out[13]=4/3 - (8*Sqrt[2])/(9*Pi)

In[14]:=%//N
Out[14]=0.933193


hth

V.Astanoff


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