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Thread: Find the smallest constant that satisfies the inequality

  1. #1
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    Find the smallest constant that satisfies the inequality

    Find the smallest constant C that satisfies the inequality: |Re(z)|+|Im(z)|\leq C|z| where z is any complex number.

    I know that |Re(z)|+|Im(z)|\leq 2|z|, but the answer is C = \sqrt{2}
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  2. #2
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     z = |z|e^{i\theta} = |z|cos(\theta) + i|z|\sin(\theta) ?
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  3. #3
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    Quote Originally Posted by Mondreus View Post
    Find the smallest constant C that satisfies the inequality: |Re(z)|+|Im(z)|\leq C|z| where z is any complex number.

    I know that |Re(z)|+|Im(z)|\leq 2|z|, but the answer is C = \sqrt{2}
    Let z = x + iy
    |Re(z)| = |x|
    |Im(z)| = |y|

    Expressed in polar form,  x = r\cos \theta ; y = r\sin \theta

    Substituting into the expression and simplifying we get
     r\cos\theta + r\sin\theta \leq Cr

     \cos\theta + \sin\theta \leq C

    The greatest value of C occurs when  \theta = \frac{\pi}{4} (this can be proved using calculus - take the first derivative and find the maximum point).

    so the greatest value C can be is  \cos \frac{\pi}{4} + \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt 2

    Therefore  C = \sqrt {2}
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