Question

Find the function whose Z transform is \(e^{\frac{1}{z}}\).

(a) log(n)

(b) \(\frac{1}{n} \)

(c) \(\frac{1}{n!} \)

(d) \(\frac{1}{(n+1)!} \)

This question was addressed to me in an international level competition.

My enquiry is from Linear Difference Equations and Z topic in chapter Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

The correct answer is (c) \(\frac{1}{n!} \)

The explanation: Using the definition of Z- Transform we have \(∑_{n=0}^∞ (\frac{1}{n!} z^{-n})\). Now, expanding this we get \( 1+\frac{z^{-1}}{1} + \frac{z^{-2}}{2} + \frac{z^{-3}}{3} \)+ …………. This is nothing but the expansion of \(e^{\frac{1}{z}}\), hence the answer is \(\frac{1}{n!}.\)