Hello everyone!
Why the difference between two elliptic integrals calculated using the built-in functions (EllipticK, EllipticF) is different from the result I obtain when I perform the explicit integration of the integrand function?
In the following an example of what I mean:
Using the built-in functions:
In[1]:= EllipticK[0.00674^2]
Out[1]= 1.57081
In[2]:= EllipticF[0.6517, 0.00674^2]
Out[2]= 0.651702
In[3]:= EllipticK[0.00674^2] - EllipticF[0.6517, 0.00674^2]
Out[3]= 0.919112
Using the explicit integration of the integrand function the complete and incomplete elliptic integrals correspond with the ones calculated with the built-in functions (neglecting the imaginary parts), but the integral between the extremes of integration corresponding to the difference of the wo integrals results different (-0.651702 instead of 0.919112):
In[4]:= f5[x_] := 1/Sqrt[1 - 0.00674^2*(Sin[x])^2]
In[5]:= Integrate[f5[x], {x, 0, Pi/2}]
Out[5]= 1.57081
In[6]:= Integrate[f5[x], {x, 0, 0.6517}]
Out[6]= 0.651702 + 3.66037*10^-18 I
In[7]:= Integrate[f5[x], {x, 0.6517, Pi/2}]
Out[7]= -0.651702 - 3.36193*10^-16 I
Where am I in wrong?
Many thanks!
