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MathGroup Archive 2007

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Re: Numerical integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73832] Re: Numerical integration
  • From: dh <dh at metrohm.ch>
  • Date: Thu, 1 Mar 2007 06:22:41 -0500 (EST)
  • References: <es3iul$obt$1@smc.vnet.net>


Hi Dimitris,

why do you want to calculate numerically an integral that can be done 

analytically? For a,b >1 we get:



-(Pi (a - b + Abs[a - b]))

--------------------------

        4 a (a - b)

Daniel



dimitris wrote:

> In another post I talk about the integral

> 

> Integrate[Cos[a x] CosIntegral[b x], {x, 0, Infinity}]

> 

> I have problems to numerical integrate this function for say

> {a,b}={3,2}.

> 

> In[20]:=

> Integrate[Cos[3*x]*CosIntegral[2*x], {x, 0, Infinity}]

> N@%

> 

> Out[20]=

> -(Pi/6)

> Out[21]=

> -0.5235987755982988

> 

> No matter how I set Options I couldn't get satisfactory results by

> NIntegrate.

> 

> Any ideas will be greatly appreciate!

> 

> Here is its plot

> 

> In[59]:=

> Plot[Cos[3*x]*CosIntegral[2*x], {x, 0, 10}, Ticks -> {Range[0, 10*Pi,

> Pi/6], Automatic}]

> 

> As we see the zeros if the function are situated at Pi/6 + n*(Pi/3),

> n=0,1,2,3...

> 

> In[61]:=

> (Cos[3*#1]*CosIntegral[2*#1] & ) /@ Table[Pi/6 + n*(Pi/3), {n, 0,

> 100}]

> 

> Out[61]=

> {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\

> 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\

> 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}

> 

> I tried to take use of this fact doing something like

> 

> In[67]:=

> lst = Table[Pi/6 + n*(Pi/3), {n, 0, 1000}] /. {a_, b__, c_} -> {x, 0,

> a, b, c};

> 

> In[70]:=

> NIntegrate[Cos[3*x]*CosIntegral[2*x], Evaluate[Sequence[lst]],

> WorkingPrecision -> 40]

> NIntegrate::ncvb :....

> -0.52359885758572151495786704

> 

> Very good result but I look for any other methods/settings!

> 

> Dimitris

> 

> 




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