Symmetries in Differential Equations

  1. It would become much easier to figure out the proof of a formula [Hob, p.26, l.-9] if we were to consider the formula and its conjugate counterpart [Hob, .26, l.-1] simultaneously. (Note. The conjugate of a complex number is a special case of symmetry.)

  2. The symmetric feature of the conditions for various values of the integral [Hob, p.360, l.14] emerges if we separate each coefficient's real part from its imaginary part [Hob, p.361, l.-9].
    Example. Compare [Hob, p.361, l.13-l.-12] with [Hob, p.363, l.-5-p.364, l.5].

  3. We may trace the reason why a theorem is symmetric by using the symmetry of conditional statements.
        Let t and t0 be variables of a theorem. If we interchange t and t0 and the theorem remains the same except for the notational difference, then we say that the theorem is symmetric with respect to t and t0 or that t and t0 are symmetric respect to the theorem.
    Example. [Har, p.54, (4.2)] is symmetric with respect to t and t0.
    Let A=(t0t) and B=(|y(t)|u0(t)).
    If we interchange t and t0 in (AB), we obtain (A'B'), where A'=(tt0) and B'=(|y(t0)|u0(t0)). The proof of [Har, p.54, (4.2)] uses (A'B') [Har, p.55, l.3], which is the symmetric counterpart of the conditional statement (AB).

    1. The symmetry between two parameters in coefficients implies the symmetry between two solutions.
          Suppose Ai(z, a, a') = Ai(z, a', a) (i = 0, 1, 2). If u(z, a) is a solution of A2(z, a, a')(d2u/dz2) + A1(z, a, a')(du/dz) + A0(z, a, a')u = f(z), so is u(z, a').
    2. Example: [Wat1, p.208, l.-16-l.-11]
    3. The symmetry between exponents for Riemann's P-equation: [Wat1, p.283, l.-6-p.284, l.6]
    4. The symmetry among regular singular points for Riemann's P-equation: [Wat1, p.284, l.7-p.285, l.-10]

  4. By symmetrizing the hypergeometric equation given in [Wat1, p.207, Example], we obtain the Riemann's P-equation [Wat1, p.206, l.-13-l.-11]. Similarly, by symmetrizing the 15 contiguous relations given in [1], we obtain the 30 contiguous relations given in [Wat1, 14.7].
    Remark. Symmetrization may generalize results, but may blur essential ideas.