Fix typo in expression for total fields from Fourier-series summation

[oskooi]

Fixes a typo in Tutorial/Cylindrical Coordinates/Nonaxisymmetric Dipole Sources involving the expression for the total fields based on the Fourier-series summation which incorrectly included taking the real part of the $m > 0$ terms.

original

modified

[stevengj]

The original looks correct to me — you take twice the real part because you are only summing positive m. Summing a positive m and a negative m term is equivalent to taking twice the real part of the positive-m term, since they are complex conjugates: z + z̄ = 2ℜ[z]

[oskooi]

If I take the Fourier-series expansion for $\vec{E}_{tot}(\theta, \phi)$ and simplify it based on $\vec{E}_m(\theta)e^{im\phi}$ being a complex conjugate for $m$ and $-m$, the resulting expression is:

[oskooi]

I have updated the formulas to:

[stevengj]

Why is that simpler? The original complex-exponential formula seems simpler to me.

[oskooi]

Why is that simpler? The original complex-exponential formula seems simpler to me.

Note that the original and the updated expressions are not equivalent. The updated expression is derived from the fact that the complex conjugate of a product of complex numbers is the product of their complex conjugates: $(ab)^* = a^* b^*$ where $a = \vec{E}_{m}(\theta)$ and $b = e^{im\theta}$.

Previously, we had been using $(ab) + (ab)^* = 2 \Re\left[ab\right]$ which is incorrect.

[mochen4]

Previously, we had been using (ab)+(ab)∗=2ℜ[ab] which is incorrect.

In fact, $(ab)+(ab)^∗=2ℜ[ab]$ looks correct to me.