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Complex Analysis : Elementary Functions

Study concepts, example questions & explanations for Complex Analysis

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Example Questions

Example Question #11 : Elementary Functions

$|e^{Z^{2}}|\leq\ e^{|Z|^{2}}$

Is the above inequality true?

Possible Answers:

Yes it is true for the whole complex plane.

No it is not true. The statement is false for the entire complex plane.

The truth of the statement depends on the value of . In other words it is true for some restricted domain of the complex plane.

It is true when:

Z=x+yi=0

The truth of the statement depends on the value of . In other words it is true for some restricted domain of the complex plane.

It is true only for the real line.

y=0

The truth of the statement depends on the value of . In other words it is true for some restricted domain of the complex plane.

It is true only for the pure imaginary line.

x=0

Correct answer:

Yes it is true for the whole complex plane.

Explanation:

$|e^{Z^{2}}|=|e^{(x+yi)^{2}}|=|e^{(x^{2}-y^{2})+(2xy)i}|$

$|e^{(x^{2}-y^{2})+(2xy)i}|=|e^{x^2-y^2}|*|e^{2xyi}|=e^{x^2-y^2}*1=e^{x^2-y^2}$

above are the steps to get the magnitude of the left side of the inequality.

$e^{|Z|^{2}}=e^{|x+yi|^{2}}=e^{(\sqrt{x^2+y^2})^{2}}=e^{x^2+y^2}$

above are the steps to get the magnitude of the right side of the inequality.

Thus $|e^{Z^{2}}|\leq\ e^{|Z|^{2}}$ becomes...

$e^{x^2-y^2}\leq e^{x^2+y^2}$

now we do algebra...

$e^{x^{2}}/e^{y^{2}}\leq e^{x^{2}}*e^{y^{2}}$

$e^{x^{2}}\leq e^{x^{2}}*e^{y^{2}}*e^{y^{2}}$

$1\leq e^{y^{2}}*e^{y^{2}}$

$1\leq e^{2y^{2}}$

with this last inequality you can graph the right hand side and see that it is

always or greater, or you can reason it this way...

$y^{2}\geqslant0$

$2y^{2}\geqslant0$

$e^{2y^{2}}\geq e^{0}$

$e^{2y^{2}}\geq 1$

Thus the inequality is true for all complex numbers.

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Example Question #12 : Elementary Functions

Compute $e^{\left(\frac{2+\pi i}{4}\right)}$

Possible Answers:

$\frac{\sqrt{2}}{2}(1+i)$

$\sqrt{2e}(1+i)$

$\sqrt{\frac{e}{2}} (1+i)$

$\sqrt{e}(1+i)$

None of the choices

Correct answer:

$\sqrt{\frac{e}{2}} (1+i)$

Explanation:

$e^{\frac{2+\pi i}{4}} = e^{\frac{2}{4}}*e^{\frac{\pi i}{4}} = \sqrt{e} * \frac{\sqrt{2}}{2}(1+i) = \sqrt{\frac{e}{2}}(1+i)$

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Example Question #13 : Elementary Functions

Compute Log(1-i)

Possible Answers:

$\frac{1}{2}\ln2 + \frac{\pi}{4}i$

$\frac{1}{2}\ln2 - \frac{\pi}{4}i$

$-\frac{i}{2}\ln2 + \frac{\pi}{4}$

$\sqrt{2} - \frac{\pi}{4}i$

$-\frac{i}{2}\ln2 - \frac{\pi}{4}$

Correct answer:

$\frac{1}{2}\ln2 - \frac{\pi}{4}i$

Explanation:

$w = \Log(1-i)$ where is the complex number such that e^w = 1-i

Converting into polar coordinates

$e^w = 1-i \implies e^xe^{iy} = \sqrt{2}e^{-\frac{\pi}{4}i} \implies x = \frac{1}{2}\ln 2 \text{ and } y = -\frac{\pi}{4}$

This gives us

$w = x+yi = \frac{1}{2}\ln2 - \frac{\pi}{4}i$

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Complex Analysis : Elementary Functions

Study concepts, example questions & explanations for Complex Analysis

All Complex Analysis Resources

Example Questions

Example Question #11 : Elementary Functions

Example Question #12 : Elementary Functions

Example Question #13 : Elementary Functions

All Complex Analysis Resources

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