How can I rederive this Mathematica result for a complex equation?

[thatboi]

Hey all,
I am trying to solve the following equation: $$\left(k-i\frac{m_{1}}{v}\right)^{-1}\Bigg(-i\frac{x}{v}-\frac{\sqrt{-x^2+m_{1}^2+k^2v^2}}{v}\Bigg) = \left(k+i\frac{m_{2}}{v}\right)^{-1}\Bigg(i\frac{x}{v}+\frac{\sqrt{-x^2+m_{2}^2+k^2v^2}}{v}\Bigg)$$ for $x$ and we can assume $m_{1},m_{2},k,v$ are all real and $i$ is the imaginary unit. Now mathematica tells me that the solution is $$x=\pm\frac{kv(m_{1}+m_{2})}{\sqrt{(m_{1}-m_{2})^2+4k^2v^2}}$$ but I am having trouble rederiving the result. Any advice would be appreciated!

 

[topsquark]

Hey all,
I am trying to solve the following equation: $$\left(k-i\frac{m_{1}}{v}\right)^{-1}\Bigg(-i\frac{x}{v}-\frac{\sqrt{-x^2+m_{1}^2+k^2v^2}}{v}\Bigg) = \left(k+i\frac{m_{2}}{v}\right)^{-1}\Bigg(i\frac{x}{v}+\frac{\sqrt{-x^2+m_{2}^2+k^2v^2}}{v}\Bigg)$$ for $x$ and we can assume $m_{1},m_{2},k,v$ are all real and $i$ is the imaginary unit. Now mathematica tells me that the solution is $$x=\pm\frac{kv(m_{1}+m_{2})}{\sqrt{(m_{1}-m_{2})^2+4k^2v^2}}$$ but I am having trouble rederiving the result. Any advice would be appreciated!

Without seeing your work there is little we can do to help.

Here's step 1 (as I see it.) Multiply both sides out and equate real and imaginary coefficients.

-Dan

 

[fresh_42]

$$\left(k-i\frac{m_{1}}{v}\right)^{-1}\Bigg(-i\frac{x}{v}-\frac{\sqrt{-x^2+m_{1}^2+k^2v^2}}{v}\Bigg) = \left(k+i\frac{m_{2}}{v}\right)^{-1}\Bigg(i\frac{x}{v}+\frac{\sqrt{-x^2+m_{2}^2+k^2v^2}}{v}\Bigg)$$
\begin{align*}
\dfrac{kv+i m_2}{kv-i m_1}&=-\dfrac{ix+\sqrt{-x^2+m_{2}^2+k^2v^2}}{ix+\sqrt{-x^2+m_{1}^2+k^2v^2}}=
-\dfrac{ix+\sqrt{-x^2+m_{2}^2+k^2v^2}}{ix+\sqrt{-x^2+m_{1}^2+k^2v^2}}\cdot \dfrac{ix-\sqrt{-x^2+m_{1}^2+k^2v^2}}{ix-\sqrt{-x^2+m_{1}^2+k^2v^2}}\\
&=\dfrac{\left(ix+\sqrt{-x^2+m_{2}^2+k^2v^2}\right)\cdot \left(ix-\sqrt{-x^2+m_{1}^2+k^2v^2}\right)}{m_1^2+k^2v^2}
\end{align*}
\begin{align*}
\left(kv+i m_2\right)\left(kv+i m_1\right)&=\left(ix+\sqrt{-x^2+m_{2}^2+k^2v^2}\right)\cdot \left(ix-\sqrt{-x^2+m_{1}^2+k^2v^2}\right)\\
&\vdots\\
&\text{etc.}
\end{align*}
... until you end up with $1=1$ or $1=0$. If it was wrong, then very likely due to a confusion of $m_1$ and $m_2$ or a sign error.