set mpi=139.57018 mmu=105.6583745 me=.5109989461  !all in MeV
set md=1232 mnu=1e-6 mp=938.272081
let Emu=(mpi^2+mmu^2)/(2*mpi)
    109.7782035074206        !MeV
let dEmu0= (-mnu^2)/(2*mpi)
  -.3582427134506813E-14     !MeV
? Emu-mmu
    4.119829007420606    !KE in MeV
? sqrt(2*res/mmu)
    .2792560758069286    !beta assuming KE/c^2=.5 m beta^2
let gamma=Emu/mmu
? sqrt(1-1/gamma^2)      !actuall beta
    .2713833487953330    

what is the maximum electron energy and velocity
electron recoil against two neutrinos which together are null
let Ee=(mmu^2+me^2)/(2*mmu)
    52.83042293020117    
let gamma=Ee/me
? sqrt(1-1/gamma^2)
    .9999532208855187    

what is the minimum electron energy and velocity
KE=0, E=me c^2; electron created at rest, neutrinos back-to-back
? (mmu-me)/2
    52.57368777695000    

lambda'=lambda+ h/me/c(1-cos(theta)), theta=180
lambda'=lambda+ 2 h/me/c

* ? 2*h/(me*c)
    .4852620472185240E-11
* ? res+500e-9
    .5000048526204722E-06

? (md^2-mp^2-mpi^2)/(2*mp)
   329.3232738056378    ! Epi MeV
? (md^2-mp^2)/(2*mp)
   339.7039701621100    !E gamma MeV