# System of equations with complex numbers

This might seem quite trivial for people who are knowledgeable in complex analysis, but it is not so much to me.

I am trying to find an efficient way to solve the following system of equations: $$\frac{11}{10} = c_1(1+i)^{-2} + \overline{c}_1(1-i)^{-2} \\ \frac{1}{2} = c_1(1+i)^{-3} + \overline{c}_1(1-i)^{-3}$$

The unknowns are obviously $c_1$ and its conjugate $\overline{c}_1$. I have tried multiplying the first equality by $(1+i)^{-1}$, substract it from the second one and so on, but it gets quite messy. I was hoping somebody could tell me a better approach.

Thanks very much

• Note that $(1+i)^{-2}=-\frac{i}{2}$ and $(1-i)^{-2}=\frac{i}{2}$. Similarly $(1+i)^{-3}=-\frac{1+i}{4}$ and $(1-i)^{-3}=\frac{1-i}{4}$. – Servaes Jun 1 '14 at 19:25
• Hint: $\overline{(1+i)}=1-i$ and $\overline{(ab)}=\overline{a}\cdot\overline{b}.$ – M. Strochyk Jun 1 '14 at 19:26
• I know the feeling. Complex numbers are both fascinating and complicated to understand. – Robert Soupe Jun 1 '14 at 19:26
• Maybe write $c_1 = x+iy$ and solve for $x,y$? – copper.hat Jun 1 '14 at 19:27

Let's write

$$z = \frac{c_1}{(1+i)^2}.$$

Then we have the equations

$$\frac{11}{10} = z + \overline{z} = 2\operatorname{Re} z$$

and

$$\frac{1}{2} = \frac{z}{1+i} + \overline{\left(\frac{z}{1+i}\right)} = 2\operatorname{Re} \frac{z}{1+i}.$$

Now note that

$$\frac{z}{1+i} = \frac{z(1-i)}{2}$$

to rewrite the second equation to

$$\frac{1}{2} = \operatorname{Re} \bigl( z(1-i)\bigr) = \operatorname{Re} z + \operatorname{Im} z.$$

From that it is not hard to determine $z$, and thence $c_1 = z(1+i)^2 = 2i z$.

Maybe it's less messy if you set $z=c_1$ and $w=1+i$. Then your equations are $$\begin{cases} w^{-2}z+\bar{w}^{-2}\bar{z}=a\\ w^{-3}z+\bar{w}^{-3}\bar{z}=b \end{cases}$$ where $a=11/10$ and $b=1/2$. The idea of multiplying the first equation by $\bar{w}^{-1}$ and subtracting is good, because you get $$w^{-2}(\bar{w}^{-1}-w^{-1})z=a\bar{w}^{-1}-b$$ so $$z=w^2\frac{a\bar{w}^{-1}-b}{\bar{w}^{-1}-w^{-1}}= w^2\frac{a-b\bar{w}}{\bar{w}}\frac{w\bar{w}}{w-\bar{w}}= w^3\frac{a-b\bar{w}}{w-\bar{w}}$$ This is just computation.

Here is another approach:

Let $c_1 = x+iy$, and substitute into the equations above. After a little tedious manipulation, this gives ${11 \over 10} = y$ and ${1 \over 2} = {y-x \over 2}$.

We can read off the solution as $c_1 = {1 \over 10}+i{11 \over 10}$.