- #1

- 10

- 0

I have an expression

##\mathcal{Im}[RT^*e^{-2ip}]=|T|^2\sin p ##, where ##R=Ae^{ip}+Be^{-ip} ## and ##p ## is a real number.

This ultimately should lead to ##\mathcal{Im}[A+B+Te^{2ip}]=0 ## upto a sign (perhaps if I didn't do a mistake).

There is a condition on ##R ## that it is real, i.e., ##R^*=R ##, but ##A ## and ##B ## are not in general real. Further, ##T## depends on ##A## and ##B ## in such a way that if ##A=0 ##, ##B=0 ## then ##T=0 ##, and ##A\neq B## so the (desired) equality holds. Here is what I do to achieve the desired result:

##\mathcal{Im}[(Ae^{ip}+Be^{-ip})T^*e^{-2ip}-|T|^2e^{ip}]=0 ##

Then I take common ##T^*e^{-ip} ## from the above expression and it leads me to

##\mathcal{Im}[\{(Ae^{2ip}+B)e^{-ip}-Te^{2ip}\}]=0 ##

This leads to ##\mathcal{Im}[Ae^{ip}+Be^{-ip}-Te^{2ip}]=0 ##

This I rewrite as (since ##R=Ae^{ip}+Be^{-ip} ## is real)

##\mathcal{Im}[A+B-Te^{2ip}]=0 ##, This is the result which is correct upto a sign.

I want to know whether I made a mistake? or there is a mistake in what I want to achieve (regarding the plus sign in front of $T$ expression in the desired versus achieved)? Thanks.

##\mathcal{Im}[RT^*e^{-2ip}]=|T|^2\sin p ##, where ##R=Ae^{ip}+Be^{-ip} ## and ##p ## is a real number.

This ultimately should lead to ##\mathcal{Im}[A+B+Te^{2ip}]=0 ## upto a sign (perhaps if I didn't do a mistake).

There is a condition on ##R ## that it is real, i.e., ##R^*=R ##, but ##A ## and ##B ## are not in general real. Further, ##T## depends on ##A## and ##B ## in such a way that if ##A=0 ##, ##B=0 ## then ##T=0 ##, and ##A\neq B## so the (desired) equality holds. Here is what I do to achieve the desired result:

##\mathcal{Im}[(Ae^{ip}+Be^{-ip})T^*e^{-2ip}-|T|^2e^{ip}]=0 ##

Then I take common ##T^*e^{-ip} ## from the above expression and it leads me to

##\mathcal{Im}[\{(Ae^{2ip}+B)e^{-ip}-Te^{2ip}\}]=0 ##

This leads to ##\mathcal{Im}[Ae^{ip}+Be^{-ip}-Te^{2ip}]=0 ##

This I rewrite as (since ##R=Ae^{ip}+Be^{-ip} ## is real)

##\mathcal{Im}[A+B-Te^{2ip}]=0 ##, This is the result which is correct upto a sign.

I want to know whether I made a mistake? or there is a mistake in what I want to achieve (regarding the plus sign in front of $T$ expression in the desired versus achieved)? Thanks.

Last edited by a moderator: