find the sum of besselh function
in [Matlab]
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From: Nathan Zhang on 17 Apr 2010 22:11 I need to find a sum of besselh(nu,K,Z) where nu from -inf to +inf. I typed symsum(besselh(n,2,beta*a),n,-inf,+inf) But encountered an error: Arguments must be numeric. Thus, I have no idea how to find this sum, could anyone help me out?? Thanks!
From: Walter Roberson on 17 Apr 2010 23:47 Nathan Zhang wrote: > I need to find a sum of besselh(nu,K,Z) where nu from -inf to +inf. > > I typed symsum(besselh(n,2,beta*a),n,-inf,+inf) > > But encountered an error: > Arguments must be numeric. > > Thus, I have no idea how to find this sum, could anyone help me out?? I believe that the symbolic functions use infinity instead of inf for infinity. The following might work: symsum('besselh(n,2,beta*a),n,-infinity,infinity)') The Maple equivalent would be, sum(HankelH2(n,beta*a),n=-infinity..infinity); but Maple does not know how to compute this for symbolic beta and a. On the other hand, if you examine the documentation for besselh, http://www.mathworks.com/access/helpdesk/help/techdoc/ref/besselh.html in the definitions section, specifies, "where ν is a nonnegative constant" and by matching the z's in the definitions, we can see that ν must correspond to the first parameter, nu, which corresponds to your parameter n. It is thus invalid to sum over n from -infinity to +infinity. This suggests that possibly your parameters are in wrong order, and that you want symsum('besselh(n,2,beta*a),n,-infinity,infinity)') This is reinforced by the second sentence of the Definitions section, "Jν(z) and Jâν(z) form a fundamental set of solutions of Bessel's equation for noninteger ν" -- but if your nu (n) were to start from -infinity, it would be indeterminate as to whether that were an integer or not (infinity is not necessarily an integer, and not necessarily _not_ an integer either!) I experimented with various numeric values, and found that Maple is unable to find a closed form for expressions such as sum(HankelH2(n+1/100,300),n=0..infinity) but it is able to evaluate them numerically. However, as that 300 is reduced to (say) 30, Maple starts refusing to evaluate it numerically, as it knows the numbers involved are too large to evaluate with the default Digits setting. (Hmmm, seems to be more complicated than that... still trying to figure out why it starts refusing.) Anyhow, I still suspect you have the parameters in the wrong order, or have the wrong limits, or some other such problem.
From: Nathan Zhang on 18 Apr 2010 22:41 Thanks for your reply so much. I tired to use infinity instead of inf, but nothing changed. I think the problem is the "nu" must be a number can't be symbolic in besselh(nu,K,Z). The interval is indeed from -infinity to +infinity. I still don't know how to find the sum. Appreciate for your help. Walter Roberson <roberson(a)hushmail.com> wrote in message <hqdvd1$2bg$1(a)canopus.cc.umanitoba.ca>... > Nathan Zhang wrote: > > I need to find a sum of besselh(nu,K,Z) where nu from -inf to +inf. > > > > I typed symsum(besselh(n,2,beta*a),n,-inf,+inf) > > > > But encountered an error: > > Arguments must be numeric. > > > > Thus, I have no idea how to find this sum, could anyone help me out?? > > I believe that the symbolic functions use infinity instead of inf for > infinity. The following might work: > > symsum('besselh(n,2,beta*a),n,-infinity,infinity)') > > > The Maple equivalent would be, > > sum(HankelH2(n,beta*a),n=-infinity..infinity); > > but Maple does not know how to compute this for symbolic beta and a. > > > On the other hand, if you examine the documentation for besselh, > http://www.mathworks.com/access/helpdesk/help/techdoc/ref/besselh.html > in the definitions section, specifies, > "where ν is a nonnegative constant" and by matching the z's in the > definitions, we can see that ν must correspond to the first parameter, > nu, which corresponds to your parameter n. It is thus invalid to sum > over n from -infinity to +infinity. This suggests that possibly your > parameters are in wrong order, and that you want > > symsum('besselh(n,2,beta*a),n,-infinity,infinity)') > > This is reinforced by the second sentence of the Definitions section, > "Jν(z) and J–ν(z) form a fundamental set of solutions of Bessel's > equation for noninteger ν" -- but if your nu (n) were to start from > -infinity, it would be indeterminate as to whether that were an integer > or not (infinity is not necessarily an integer, and not necessarily > _not_ an integer either!) > > I experimented with various numeric values, and found that Maple is > unable to find a closed form for expressions such as > > sum(HankelH2(n+1/100,300),n=0..infinity) > > but it is able to evaluate them numerically. However, as that 300 is > reduced to (say) 30, Maple starts refusing to evaluate it numerically, > as it knows the numbers involved are too large to evaluate with the > default Digits setting. (Hmmm, seems to be more complicated than that... > still trying to figure out why it starts refusing.) > > Anyhow, I still suspect you have the parameters in the wrong order, or > have the wrong limits, or some other such problem.
From: Walter Roberson on 19 Apr 2010 00:06 Nathan Zhang wrote: > I tired to use infinity instead of inf, but nothing changed. I think the > problem is the "nu" must be a number can't be symbolic in besselh(nu,K,Z). > The interval is indeed from -infinity to +infinity. > I still don't know how to find the sum. Appreciate for your help. Well, you are not going to be able to find the sum you are looking for, as the besselh function is defined in terms of non-negative non-integral nu. Looking at the definitions of the Hankel functions, I always see nu^2, which suggests that the sum from -infinity to +infinity should be twice the sum from 0 to infinity. Maple permits the nu argument to HankelH2 to be negative or integral, but it does not appear to have any means to symbolically integrate it.
From: Nathan Zhang on 19 Apr 2010 01:11 Thanks for your instant reply. Even the value of "nu" is from 0 to +infinity as integer, how to find the sum?? Since I can't use a symbolic letter "n" to set it into "nu" in the function besselh(), I still have no idea to solve the problem. Thank you so much.
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