@Axel Vogt 

@Axel Vogt 

Thank you very much for your reply! In your opinion, can the curves vs afa be not smooth?

@vv 

I believe the curves should be smooth, because for some other parameters, the curves are all smooth. Maybe the curves are not smooth for the present parameters?

@vv 

The integrand may be not continuous, but should be analytical. Any way, the curves should be smooth.

@Axel Vogt 

Thank you very much for your reply. There are, in fact, singularities, because I have added a small imaginary part for gama. Mathematica or Matlab can give a finite result. However, the curves given by Mathematica or Matlab are not smooth (In principle, the integrand is continuous and analytical, and the curves of the integration should be smooth.). This is why I turn to use Maple to handle the problem, and I hope there exists good and reliable methods to treat such problem in Maple.

@vv 

Thank you for your reply. However, I have increased the Digits, it does not work. I have optimize the computations, but it may be still complicated in mathematics.

@vv 

Thank you very much for your reply. From other parameters, I found that when the precision increases the results can be much different. However, for afa=0.277, when the precision is set as epsilon=1e-4, the program can not work in half an hour. What is wrong? How to solve this problem?

@vv 

Thank you very much for your answer! I am not very clear that what does your modification mean. Can you tell me in detail what is wrong about my original program? I understand that you set T2 a value of x=Pi/4 and y=afa, however, T2 should be a function of x and y. Therefore, your answer may give a wrong result. I am looking forward to hearing from you soon. Thank you very much!

T2.mw

I have modified some parameters in the previous program, and the new program "T2.mw" is attached above. When the "afa" is larger than 0.277, the program say:"The integral is probably divergent or slowly convergent". However, the integral can be worked out by Mathematica. If the integral is  divergent, why Mathematica can get a result? Thank you very much!

@Axel Vogt 

@Axel Vogt 

Thank you very much for your answer!  Your answer is very important and I believe you have got the point. However, I am not quite familiar with Maple, and I do not know how to run the program you give. Can you tell me how to run the program you give?