# Convergence Problem of an exponential

I want to evaluate the following integral numerically

NIntegrate[w E^(-w/0.001)/(w(w-0.0001))^2), {w,0, 0.0001, 1},
MaxRecursion -> 300, AccuracyGoal -> 10, Method -> "PrincipalValue"]


It gives the error

Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small

It seems that a lot of questions arose regarding this problem in the community, but I didn't find any concrete solution. I really appropriate if anyone help me out with a solution?

• It seems that w=0 is another location of singularity? – tablecircle Nov 13 '16 at 5:58
• @tablecircle Yeah. But I think that the principal value method takes care of that. – New Developer Nov 13 '16 at 6:42

Your Integral doesn't converge at all, it is Infinity, therefore the error message. As you see, the Denominator ist 0 at w=0 and at w=.0001. Make a logarithmic Plot to see it. Next hint: often it is better to use rationalized values instead of real numbers. Principalvalue only makes sense, if a positive part at the singularity is compensated by an equal large negative part. You have only positive parts at 0 and .0001.

       In[60]:= LogPlot[w E^(-w/0.001)/(w (w - 0.0001))^2, {w, 0, .0002},
MaxRecursion -> 15]

In[9]:= w E^(-w 1000)/(w (w - 1/10000))^2

Out[9]= E^(-1000 w)/((-(1/10000) + w)^2 w)

In[38]:= Plot[((-(1/10000) + w)^2 w), {w, 0, 2/10000}]

In[27]:= sol = Solve[(-(1/10000) + w)^2 w == 0, w]

Out[27]= {{w -> 0}, {w -> 1/10000}, {w -> 1/10000}}

In[29]:= E^(-1000 w) /. sol // N

Out[29]= {1., 0.904837, 0.904837}

In[35]:= 1/((-(1/10000) + w)^2 w) // ExpandAll

Out[35]= 1/(w/100000000 - w^2/5000 + w^3)

In[57]:= Integrate[w E^(-w 1000)/(w (w - 1/10000))^2, {w, 0,   1/100000}]

During evaluation of In[57]:= Integrate::idiv: Integral of      E^(-1000 w)/((-(1/10000)+w)^2 w) does not converge on {0,1/100000}. >>

Out[57]= \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$0$$,
FractionBox[$$1$$, $$100000$$]]$$\*FractionBox[ SuperscriptBox[\(E$$, $$\(-1000$$\ w\)], $$\*SuperscriptBox[\((\(- \*FractionBox[\(1$$, $$10000$$]\) +
w)\), $$2$$]\ w\)] \[DifferentialD]w\)\)

In[58]:= Integrate[w E^(-w 1000)/(w (w - 1/10000))^2, {w,   1/100000, 1}]

During evaluation of In[58]:= Integrate::idiv: Integral of      E^(-1000 w)/((-(1/10000)+w)^2 w) does not converge on {1/100000,1}. >>

Out[58]= \!$$\*SubsuperscriptBox[\(\[Integral]$$,
FractionBox[$$1$$, $$100000$$], $$1$$]$$\*FractionBox[ SuperscriptBox[\(E$$, $$\(-1000$$\ w\)], $$\*SuperscriptBox[\((\(- \*FractionBox[\(1$$, $$10000$$]\) +
w)\), $$2$$]\ w\)] \[DifferentialD]w\)\)

In[64]:= Plot[
w E^(-w 1000)/(w (w - 1/10000))^2, {w, -1/100000, 1/100000}]

In[65]:= NIntegrate[
w E^(-w 1000)/(w (w - 1/10000))^2, {w, -1/100000, 0, 1/100000},
Method -> "PrincipalValue"]

Out[65]= 3.82496*10^7