 #1
 1,266
 11
Homework Statement:
 What is the correct way of computing the argument of the following equation?
Relevant Equations:

I am trying to compute the argument ##\Phi## of the equation
$$\frac{r\tau\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)} \tag{1}$$
which using Euler's equation can also be written in the form
$$\exp\left[i\left(\pi+\varphi\right)\right]\frac{\taur\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)} \tag{2}$$
Problem Statement: What is the correct way of computing the argument of the following equation?
Relevant Equations: I am trying to compute the argument ##\Phi## of the equation
$$\frac{r\tau\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)} \tag{1}$$
which using Euler's equation can also be written in the form
$$\exp\left[i\left(\pi+\varphi\right)\right]\frac{\taur\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)} \tag{2}$$
(1) If we use the first equation, we can first separate out the real and imaginary parts of the expression by multiplying by the complex conjugate of the denominator
$$\frac{r\tau\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)}.\frac{1\tau r\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)}=\frac{r\tau r^{2}\exp\left(i\varphi\right)\tau\exp\left(i\varphi\right)+\tau^{2}r}{1\tau r\left[\exp\left(i\varphi\right)+\exp\left(i\varphi\right)\right]+\left(\tau r\right)^{2}}$$
$$=\frac{r+\tau^{2}r\tau\left(r^{2}+1\right)\cos\varphi}{1\tau r\cos\varphi+\left(\tau r\right)^{2}}+i\frac{\tau\left(1r^{2}\right)\sin\varphi}{1\tau r\cos\varphi+\left(\tau r\right)^{2}}.$$
Since, for a complex number ##z##, ##\text{arg}\left(z\right)=\text{atan }\left[\Im\left(z\right)/\Re\left(z\right)\right]##, we have:
$$\Phi=\text{atan}\left[\frac{\tau\left(1r^{2}\right)\sin\varphi}{r+\tau^{2}r\tau\left(r^{2}+1\right)\cos\varphi}\right].$$
(2) However, the paper I am looking at used the second form (equation (2)), which readily gives:
$$\Phi=\pi+\varphi+\text{atan}\left(\frac{r\sin\varphi}{\taur\cos\varphi}\right)+\text{atan}\left(\frac{r\tau\sin\varphi}{1r\tau\cos\varphi}\right).$$
Clearly, these two answers are very different. Which method is correct, and what is the cause of the discrepancy? Shouldn't we end up with the same expression for the argument regardless of the form we start with?
Any explanation is appreciated.
Relevant Equations: I am trying to compute the argument ##\Phi## of the equation
$$\frac{r\tau\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)} \tag{1}$$
which using Euler's equation can also be written in the form
$$\exp\left[i\left(\pi+\varphi\right)\right]\frac{\taur\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)} \tag{2}$$
(1) If we use the first equation, we can first separate out the real and imaginary parts of the expression by multiplying by the complex conjugate of the denominator
$$\frac{r\tau\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)}.\frac{1\tau r\exp\left(i\varphi\right)}{1\tau r\exp\left(i\varphi\right)}=\frac{r\tau r^{2}\exp\left(i\varphi\right)\tau\exp\left(i\varphi\right)+\tau^{2}r}{1\tau r\left[\exp\left(i\varphi\right)+\exp\left(i\varphi\right)\right]+\left(\tau r\right)^{2}}$$
$$=\frac{r+\tau^{2}r\tau\left(r^{2}+1\right)\cos\varphi}{1\tau r\cos\varphi+\left(\tau r\right)^{2}}+i\frac{\tau\left(1r^{2}\right)\sin\varphi}{1\tau r\cos\varphi+\left(\tau r\right)^{2}}.$$
Since, for a complex number ##z##, ##\text{arg}\left(z\right)=\text{atan }\left[\Im\left(z\right)/\Re\left(z\right)\right]##, we have:
$$\Phi=\text{atan}\left[\frac{\tau\left(1r^{2}\right)\sin\varphi}{r+\tau^{2}r\tau\left(r^{2}+1\right)\cos\varphi}\right].$$
(2) However, the paper I am looking at used the second form (equation (2)), which readily gives:
$$\Phi=\pi+\varphi+\text{atan}\left(\frac{r\sin\varphi}{\taur\cos\varphi}\right)+\text{atan}\left(\frac{r\tau\sin\varphi}{1r\tau\cos\varphi}\right).$$
Clearly, these two answers are very different. Which method is correct, and what is the cause of the discrepancy? Shouldn't we end up with the same expression for the argument regardless of the form we start with?
Any explanation is appreciated.