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To represent a simple sinusoidally varying function $V(t)$ let us use $V(t)=Re(\hat V e^{i\omega t})$ where $\hat V$ can be a complex constant.Let $\hat V =-iV_o$ Therefore, $V(t)=V_o \sin(\omega t)$. Similarly, how can I represent the following function in terms of a complex function whose real value yields the function mentioned below:- $$Y(t)=|Y_0\sin(\omega t)|$$ To account for the the fact that both the half of the sinusoidal cycle will be positive, I first tried to represent $Y(t)=A\sin^2 (Ct)$ for some $A$ and $C$ and then use $\sin^2x=1-\frac{1+\cos(2x)}{2}$ to transform it into the required complex function.

But since the mathematical characteristics of the squared sine function is different from the absolute value of sine function (though the graphs have a similar appearance), the method doesn't yield the required function. How then, if possible to convert $Y(t)$ to a complex function?

Can we use Fourier series here to get the function as a sum of sines and cosines? I know:- $$|\sin(\omega t)|=-\sin(\omega t)\text{ for }\sin(\omega t)<0\text{ and } =\sin(\omega t)\text{ for }\sin(\omega t)>0$$

[I have used the physics tag because in AC circuit analysis, we characterize the varying impedances, currents and the voltages as complex numbers (exponentials). But at all times, the real situation is depicted by only the real part of complex currents. The exponentials are reduced by using $e^{i\theta}=\cos{\theta}+i\sin{\theta}$ into the real part which varies sinusoidally. Using a full wave recitifier we get $V_{in}(t)=V_0\sin\omega t$ but $V_{out}(t)=V_0|\sin\omega t|$]

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$$ Y(t) = |\sin (\omega t) | \\ $$ can be extended to the complex plain using the formula $Im(z) = \dfrac{z - z^*}{2i}$. From this prove that $\sin : \Bbb{C} \to \Bbb{C}$, defined by

$$ \sin(z) = \frac{e^{iz} - e^{-iz}}{2i} $$

is an extension of $\sin : \Bbb{R} \to \Bbb{R}$. Then $Y : \Bbb{C} \to \Bbb{C}$ using this definition is $Y(t) = \left |\dfrac{e^{i\omega t} - e^{-i \omega t}}{2i} \right |$, where $t \in \Bbb{C}$. But $|z| = \sqrt{z z^*}$ in $\Bbb{C}$. So you could expand that further if you wanted.

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  • $\begingroup$ can we get an expression for $Y(t)$ without using the absolute value function(so that it can be simply analysed)? $\endgroup$ – Satwik Pasani Nov 15 '13 at 2:22
  • $\begingroup$ Like I said $|z| = \sqrt{zz^*}$ for all $z \in \Bbb{C}$. $\endgroup$ – Shine On You Crazy Diamond Nov 18 '13 at 15:55
  • $\begingroup$ So $z= \left |\dfrac{e^{i\omega t} - e^{-i \omega t}}{2i} \right |$ and $ z*= \left |\dfrac{e^{i\omega t} + e^{-i \omega t}}{2i} \right |$ ???? $\endgroup$ – Nick Dec 13 '15 at 0:20

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