Many thanks to you both for the solutions. 
Not surprised Mathematica simplifies this straightforwardly. 

@acer 

Very interesting. I am using Maple16 on a macbook and it produces exactly the same output as you have. 

Why does the line defining hmm not produce the imaginary part? I was excited at seeing this thinking the substitution was not required.

The result produced ifor hmm is the same as the real part of the analytical solution I quoted earlier. I believe the result I quoted is valid only for alpha>0 and beta>0. 

@Ian Jones 

The substitution suggested by Maxim also appears to work for the zero-th order functions 

restart:
gr:=(beta,r)->(1/8*I)*(-HankelH1(0, beta*r)-(2*I)*BesselK(0, beta*r)/Pi)/beta^2;
p:=subs(s = sqrt(tau), (1/2)*tau^(-1/2)*convert(cos(alpha*s)*gr(beta, s), MeijerG, include = cos));   
q:=int(p,tau=0..infinity);
assume(alpha>0);assume(beta>0);
simplify (q);

Check analytical result
simplify(q-((1/(8*beta^2))*(1/sqrt(beta^2+alpha^2)-I/sqrt(beta^2-alpha^2))));
alpha:=1;beta:=3/2;sigma:=-1/2;
evalf(q);

Interestingly, Maple evaluates the integral in general before the one applies the assumptions. Trying to help it by introducing earlier the assumptions about alpha and beta ie. move the assumption line to just above the definition of q and Maple fails to find the integral for the special case alpha>0 and beta>0.

@Markiyan Hirnyk 

Apologies. The result quoted was for the zero-the order Hankel1 and BesselK

restart;
with(plots):assume(alpha>0);assume(beta>0);
gr:=(beta,r)->(1/8*I)*(-HankelH1(0, beta*r)-(2*I)*BesselK(0, beta*r)/Pi)/beta^2;
alpha:=1;beta:=3/2;
evalf(Int(cos(alpha*s)*gr(beta,s),s=0..infinity));
evalf((1/(8*beta^2))*(1/sqrt(beta^2+alpha^2)-I/sqrt(beta^2-alpha^2)));

@vv 

It looks ok I think. The imaginary number I appears in the second term in brackets. It has been validated numerically in Mathematica by a colleague.

Hello all.

The analytical result is (1/8/beta^2)*(1/sqrt(alpha^2+beta^2)-I/sqrt(beta^2-alpha^2))  for alpha>0 and beta>0

Clearly, 

(i) the case alpha=beta blows up as observed above

(ii) if alpha>beta additionally then the integral is real.

The above result can be found DIRECTLY in Mathematica. I have used Maple as my computer algebra choice for many years but it does have drawbacks. I assume that, without the workaround using generalised functions given kindly by Maxim_372, the integral cannot be done directly in Maple. Is that correct? Is there a workaround as a special case of the hypergeometric function only (not involving MeijerG)?

Many thanks

Hello all.

The analytical result is (1/8/beta^2)*(1/sqrt(alpha^2+beta^2)-I/sqrt(beta^2-alpha^2))  for alpha>0 and beta>0

Clearly,

(i) the case alpha=beta blows up as observed above

(ii) if alpha>beta additionally then the integral is real.

The above result can be found DIRECTLY in Mathematica. I have used Maple as my computer algebra choice for many years but it does have drawbacks. I assume that, without the workaround using generalised functions given kindly by Maxim_372, the integral cannot be done directly in Maple. Is that correct? Is there a workaround as a special case of the hypergeometric function only (not involving MeijerG)?

Many thanks

@_Maxim_ 

Many thanks for this. I will look in detail.

Hi Dmitri,

I remember your great help when we were doing the online exams last year and I appreciate your positive reply.

We would hope to try and implement the connector for next academic year if this change could be implemented.

With best wishes,

Ian 

Many thanks, Jason. Hadn't picked up the point about the item stats and essay questions

Many thanks, Jason. Hadn't picked up the point about the item stats and essay questions

Hi Alex,

Thanks very much for this info. We have been working with Tech Support at Maplesoft and they are coming in to our server tomorrow. We shall await results...

It is amazing that the old system properties, such as session idle time etc, are now buried in the postgres database or in the file web.xml. These used to be easily accessible in v2.5

I'm very reluctant to change from 3.01 to 4 in mid semester. On past form, many bugs may have been corrected but there may be fresh problems with v4.

Ian 

Hi Alex,

Thanks very much for this info. We have been working with Tech Support at Maplesoft and they are coming in to our server tomorrow. We shall await results...

It is amazing that the old system properties, such as session idle time etc, are now buried in the postgres database or in the file web.xml. These used to be easily accessible in v2.5

I'm very reluctant to change from 3.01 to 4 in mid semester. On past form, many bugs may have been corrected but there may be fresh problems with v4.

Ian 

Hi Alex,

I changed all users to students and had a class today. Had a handful of students who experienced the same old problem. I checked and they were 'students'. Do you have suggestions?

Many thanks once again,

Ian

Hi Alex,

I changed all users to students and had a class today. Had a handful of students who experienced the same old problem. I checked and they were 'students'. Do you have suggestions?

Many thanks once again,

Ian