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## Homework Statement

Show that if Im(z) ≥ 0, then |[itex]{{\rm e}^{i \left( x+iy \right) }}[/itex]| ≤ 1.

Let R > 1 be a real constant

Now deduce that [itex]{\frac {{{\rm e}^{-{\it Im} \left( x+iy \right) }}}{ \left| \left( x+

iy \right) ^{4}+1 \right| }}

[/itex] ≤ [itex] \left( {R}^{4}-1 \right) ^{-1}[/itex] for z on the semi-circle {z [itex]\in[/itex] ℂ:

**|z| = R**, Im(

**z**) ≥ 0}

## The Attempt at a Solution

The imaginary part of z ≥ 0 is just y ≥ 0.

Multiplying the exponential through, you get, [itex]{{\rm e}^{ix-y}}[/itex].

I realise that taking the modulus of this will make the e^ix disappear, leaving only e^-y, which clearly shows then as y ≥ 0:

e^-y ≤ 1

I'm not too sure how to get the modulus of that exponential.

N.B: The modulus applies to the entire fraction, I cannot seem to get the modulus including the entire fraction for some reason, it's just appearing on the denominator

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