Q7Advanced Engineering Mathematics

In finite difference calculus, the foundational operators are intrinsically linked. The Central Difference operator $\delta$ is defined as $E^{1/2}-E^{-1/2}$, where $E$ is the Shift operator. Squaring this yields ${\delta }^2=E-2+E^{-1}$. By algebraically manipulating the expression $1+\frac{1}{2}{\delta }^2+\delta \sqrt{1+\frac{{\delta }^2}{4}}$ and substituting these fundamental relations, it simplifies perfectly back to the original operator $E$.