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Engineering LibreTexts

4.7: Numerical Experiment (Approximating e^jθ)

  • Page ID
    10097
  • We have demonstrated that the function \(e^{jθ}\) has two representations:

    1. \(e^{jθ}=\lim_{n→∞}(1+\frac {jθ} n)^n\)  ; and
    2. \(e^{jθ}=\lim_{n→∞}∑_{k=0}^n\frac {(jθ)^k} {k!}\)

    In this experiment, you will write a MATLAB program to evaluate the two functions \(f_n\) and \(S_n\) for twenty values of n:

    1. \(f_n=(1+\frac {jθ} n)^n,\;n=1,2,...,20\) ; and
    2. \(S_n=∑^n_{k=0}\frac {(jθ)^k} {k!},\;n=1,2,...,20k\)

    Choose \(θ=π/4(=\mathrm{pi}/4)\). Use an implicit for loop to draw and plot a circle of radius 1. Then use an implicit for loop to compute and plot \(f_n\) and an explicit for loop to compute and plot \(S_n\) for n=1,2,...,100. You should observe plots like those illustrated in the Figure. Interpret them.

    ConvergencePlots.PNG
    Plots for Convergence;(a) of fn, and (b) of Sn