Mathematica integral with many singularities

Question

What's the best way of getting Mathematica 7 or 8 to do the integral

NIntegrate[Exp[-x]/Sin[Pi x], {x, 0, 50}]

There are poles at every integer - and we want the Cauchy principle value. The idea is to get a good approximation for the integral from 0 to infinity.

With Integrate there is the option PrincipleValue -> True.

With NIntegrate I can give it the option Exclusions -> (Sin[Pi x] == 0), or manually give it the poles by

NIntegrate[Exp[-x]/Sin[Pi x], Evaluate[{x, 0, Sequence@@Range[50], 50}]]

The original command and the above two NIntegrate tricks give the result 60980 +/- 10. But they all spit out errors. What is the best way of getting a quick reliable result for this integral without Mathematica wanting to give errors?

Answer 1

Simon, is there reason to believe your integral is convergent ?

In[52]:= f[k_Integer, eps_Real] := 
 NIntegrate[Exp[-x]/Sin[Pi x], {x, k + eps, k + 1 - eps}]

In[53]:= Sum[f[k, 1.0*10^-4], {k, 0, 50}]

Out[53]= 2.72613

In[54]:= Sum[f[k, 1.0*10^-5], {k, 0, 50}]

Out[54]= 3.45906

In[55]:= Sum[f[k, 1.0*10^-6], {k, 0, 50}]

Out[55]= 4.19199

It looks like the problem is at x==0. Splitting integrand k+eps to k+1-eps for integer values of k:

In[65]:= int = 
 Sum[(-1)^k Exp[-k ], {k, 0, Infinity}] Integrate[
   Exp[-x]/Sin[Pi x], {x, eps, 1 - eps}, Assumptions -> 0 < eps < 1/2]

Out[65]= (1/((1 + 
   E) (I + \[Pi])))E (2 E^(-1 + eps - I eps \[Pi])
     Hypergeometric2F1[1, (I + \[Pi])/(2 \[Pi]), 3/2 + I/(2 \[Pi]), 
     E^(-2 I eps \[Pi])] + 
   2 E^(I eps (I + \[Pi]))
     Hypergeometric2F1[1, (I + \[Pi])/(2 \[Pi]), 3/2 + I/(2 \[Pi]), 
     E^(2 I eps \[Pi])])

In[73]:= N[int /. eps -> 10^-6, 20]

Out[73]= 4.1919897038160855098 + 0.*10^-20 I

In[74]:= N[int /. eps -> 10^-4, 20]

Out[74]= 2.7261330651934049862 + 0.*10^-20 I

In[75]:= N[int /. eps -> 10^-5, 20]

Out[75]= 3.4590554287709991277 + 0.*10^-20 I

As you see there is a logarithmic singularity.

In[79]:= ser = 
 Assuming[0 < eps < 1/32, FullSimplify[Series[int, {eps, 0, 1}]]]

Out[79]= SeriesData[eps, 0, {(I*(-1 + E)*Pi - 
     2*(1 + E)*HarmonicNumber[-(-I + Pi)/(2*Pi)] + 
          Log[1/(4*eps^2*Pi^2)] - 2*E*Log[2*eps*Pi])/(2*(1 + E)*Pi), 
     (-1 + E)/((1 + E)*Pi)}, 0, 2, 1]

In[80]:= Normal[
  ser] /. {{eps -> 1.*^-6}, {eps -> 0.00001}, {eps -> 0.0001}}

Out[80]= {4.191989703816426 - 7.603403526913691*^-17*I, 
 3.459055428805136 - 
     7.603403526913691*^-17*I, 
 2.726133068607085 - 7.603403526913691*^-17*I}

EDIT Out[79] of the code above gives the series expansion for eps->0, and if these two logarithmic terms get combined, we get

In[7]:= ser = SeriesData[eps, 0, 
       {(I*(-1 + E)*Pi - 2*(1 + E)*HarmonicNumber[-(-I + Pi)/(2*Pi)] + 
              Log[1/(4*eps^2*Pi^2)] - 2*E*Log[2*eps*Pi])/(2*(1 + E)*
       Pi), 
         (-1 + E)/((1 + E)*Pi)}, 0, 2, 1]; 

In[8]:= Collect[Normal[PowerExpand //@ (ser + O[eps])], 
 Log[eps], FullSimplify]

Out[8]= -(Log[eps]/\[Pi]) + (
 I (-1 + E) \[Pi] - 
  2 (1 + E) (HarmonicNumber[-((-I + \[Pi])/(2 \[Pi]))] + 
     Log[2 \[Pi]]))/(2 (1 + E) \[Pi])

Clearly the -Log[eps]/Pi came from the pole at x==0. So if one subtracts this, just like principle value method does this for other poles you end up with a finitely value:

In[9]:= % /. Log[eps] -> 0

Out[9]= (I (-1 + E) \[Pi] - 
 2 (1 + E) (HarmonicNumber[-((-I + \[Pi])/(2 \[Pi]))] + 
    Log[2 \[Pi]]))/(2 (1 + E) \[Pi])

In[10]:= N[%, 20]

Out[10]= -0.20562403655659928968 + 0.*10^-21 I

Of course, this result is difficult to verify numerically, but you might know more that I do about your problem.

EDIT 2

This edit is to justify In[65] input that computes the original regularized integral. We are computing

Sum[ Integrate[ Exp[-x]/Sin[Pi*x], {x, k+eps, k+1-eps}], {k, 0, Infinity}] ==  
  Sum[ Integrate[ Exp[-x-k]/Sin[Pi*(k+x)], {x, eps, 1-eps}], {k, 0, Infinity}] ==
  Sum[ (-1)^k*Exp[-k]*Integrate[ Exp[-x]/Sin[Pi*x], {x, eps, 1-eps}], 
       {k, 0, Infinity}] == 
  Sum[ (-1)^k*Exp[-k], {k, 0, Infinity}] * 
     Integrate[ Exp[-x]/Sin[Pi*x], {x, eps, 1-eps}]

In the third line Sin[Pi*(k+x)] == (-1)^k*Sin[Pi*x] for integer k was used.

Answer 2

Simon, I haven't spent much time with your integral, but you should try looking at stationary phase approximation. What you have is a smooth function (exp), and a highly oscillatory function (sine). The work involved is now in brow-beating the 1/sin(x) into the form exp(if(x))

Alternatively, you could use the series expansion of the cosecant (not valid at poles):

In[1]:=Series[Csc[x], {x, 0, 5}]
(formatted) Out[1]=1/x + x/6 + 7/360 x^3 + 31/15120 x^5 +O[x]^6

Note that for all m>-1, you have the following:

In[2]:=Integrate[x^m Exp[-x], {x, 0, Infinity}, Assumptions -> m > -1]
Out[2]=Gamma[1+m]

However, summing the series with the coefficients of cosecant (from wikipedia), not including 1/x Exp[-x] case, which doesn't converge on [0,Infinity].

c[m_] := (-1)^(m + 1) 2 (2^(2 m - 1) - 1) BernoulliB[2 m]/Factorial[2 m];
Sum[c[m] Gamma[1 + 2 m - 1], {m, 1, Infinity}]

does not converge either...

So, I'm not sure that you can work out an approximation for the integral to infinity, but I if you're satisfied with a solution upto some large N, I hope these help.