Discussion
Which of the following is not correct ?
Answer: D
The incorrect relation is \(\delta = E^{1/2} + E^{-1/2}\).
The central difference is \(\delta = E^{1/2} - E^{-1/2}\), with a minus sign. The plus combination is \(2\mu\), twice the averaging operator.
The others are correct. With \(\Delta = E - 1\) and \(\nabla = 1 - E^{-1}\): \(\Delta - \nabla = E - 2 + E^{-1} = (E - 1)(1 - E^{-1}) = \Delta\nabla\); \((1 + \Delta)(1 - \nabla) = E \cdot E^{-1} = 1\); and with \(\mu = \tfrac{1}{2}(E^{1/2} + E^{-1/2})\),
\[ \begin{aligned} \mu^{2} &= \tfrac{1}{4}(E + 2 + E^{-1}) \\ &= 1 + \tfrac{1}{4}\delta^{2} \end{aligned} \]
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