# Sum of complex numbers in exponential form formula inconsistency

Let $$z_1 = r_1 \exp(i \theta_1)$$ and $$z_2 = r_2\exp(i \theta_2)$$ be complex numbers in exponential form and $$z = z_1 + z_2$$, then I know using the complex plane vector representation that $$z$$ can be written in the exponential form too, $$z = r \exp(i \theta)$$, where $$r = \sqrt{r_1^2 + r_2^2 + 2 r_1 r_2 \cos( \theta_1 - \theta_2)}$$ $$\theta = \arctan\left(\frac{r_1 sin\theta_1 + r_2 sin\theta_2} {r_1 cos\theta_1 + r_2 cos\theta_2}\right)$$

Problem: when I plotted-in Mathematica- the real part of $$z$$, it did not match with the plot of the real part of $$z_1 + z_2$$ but rather it matched the plot of the real part of its absolute value $$|{z_1 + z_2}|$$. I tried two different numbers for $$z_1, z_2$$, both have the problem.

Question: 1) Why is that, what is wrong with the formula?, 2) How to fix it so both $$Re[z_1 + z_2]$$ and $$Re[z]$$ give the same plot?

• I don't know if this will solve your problem, but try this: the arctangent formula for the angle only works when the angle of the resulting complex number is between $-\pi/2$ and $\pi/2$. Try adjusting the whole thing by an angle of $\pi$. Jul 20, 2020 at 19:09

$$z = z_1+z_2 = r_1\cos\theta_1 + r_2\cos\theta_2 + ir_1\sin\theta_1+ir_2\sin\theta_2$$.
$$\mod(z) = \sqrt{(Re(z))^2 + (Im(z))^2} = \sqrt{r_1^2(\cos^2\theta_1 + \sin^2\theta_1) + r_2^2(\cos^2\theta_2 + \sin^2\theta_2) + 2r_1r_2(\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2)} = \sqrt{r_1^2+r_2^2+2r_1r_2\cos(\theta_1-\theta_2)}$$.
$$\arg(z) = \arctan(Im(z)/Re(z)) = \arctan(\frac{r_1\sin\theta_1 + r_2\sin\theta_2}{r_1\cos\theta_1 + r_2\cos\theta_2})$$.