Complex Analysis – Evaluate Difference of Min and Max Value of |z1^3 + z2^3|

complex numberscomplex-analysismaxima-minima

Question:

Given that,
$$|z_1 + z_2| = 20 \quad \text{and} \quad |z_1^2 + z_2^2| = 16
$$

Given these conditions, we need to find the difference between the maximum and minimum values of $|z_1^3 + z_2^3|$.

I undersdand this problem requires careful manipulation of complex numbers. But i couldn't manipulate it to a reasonable extent.

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We're asked to find the difference between the maximum and minimum values of $|z_1^3 + z_2^3|$. Using De Moivre's theorem, we have:

Given hint in the answer key:
To find the extreme values, we differentiate with respect to $\theta_1 – \theta_2$, equate it to zero, and solve for $\theta_1 – \theta_2$. We then substitute this value back to get the maximum and minimum values of $|z_1^3 + z_2^3|$.

This seems way lengthy and unreasonable, is there any elegant or better approach than this?

Best Answer

(The idea similar as in John Bentin's answer .)

With $a=z_1 + z_2$ and $b = z_1^2 + z_2^2$ is $$ z_1^3 + z_2^3 = -\frac 12 a (a^2 - 3b) $$ and therefore $$ |z_1^3 + z_2^3| = \frac 12 |a| \cdot |a^2 - 3b| = 10 |a^2 -3b| \\ \begin{cases} \le 10 (|a|^2 + 3|b|) = 4480 \\ \ge 10 (|a|^2 - 3|b|) = 3520 \end{cases} $$ by the triangle inequality. The bounds are sharp:

  • The upper bound is attained if $a^2/b$ is a negative real number, e.g. for $(a,b) = (20, -16)$.
  • The lower bound is attained if $a^2/b$ is a positive real number, e.g. for $(a, b) = (20, 16)$.

Both cases are possible because the system $$z_1+z_2=a, z_1^2+z_2^2=b$$ has solutions for any pair $(a, b)$ of complex numbers.

Therefore the desired difference is $4480-3520 = 960$.

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