Circular antenna array At page 20 of http://read.pudn.com/downloads773/doc/comm/3065154/5237234.pdf the two formulas that characterize the behavior of a linear and a circular antenna array are compared.
The reason why I write this post is: I don't understand why the second formula is true.
Below I show why I am not convinced.
In general, given a configuration of sources (currents impressed) in the vacuum we can write that the vector potential is equal to:
$$\mathbf{A}(\mathbf{r})=\frac{e^{-jkr}}{4\pi r}\underbrace{\int_V \mathbf{J}(\mathbf{r}')e^{jk\mathbf{r}_0\cdot \mathbf{r}'}\mathrm{d}V}_{\mathcal{a}(\theta,\phi)}$$
where (as usual) $k=\frac{2\pi f}{c}$ ($f$ frequency and $c$ speed of light), $\mathbf{r}_0$ unit vector pointing at the field evaluation point (at a distance $r$ from the origin) and $\mathbf{r}'$ source point.
We detail this expression for a discrete circular configuration on the xy plane, where the points drawed are the 'centers' of the current distribution of every element in the array:

and we use the cylindrical coordinates for $\mathbf{r}'$:
$$\mathcal{a}(\theta,\phi)=\int_V \mathbf{J}(r',\phi',z')e^{jk\left (r'\sin\theta\cos (\phi-\phi')+z'\cos\theta  \right )}r'\mathrm{d}r'\mathrm{d}\phi'\mathrm{d}z'$$
which can be written equivalently as:
$$\mathcal{a}(\theta,\phi)=\sum_n\int_{V_n} \mathbf{J}^{(n)}(r',\phi',z')e^{jk\left (r'\sin\theta\cos (\phi-\phi')+z'\cos\theta  \right )}r'\mathrm{d}r'\mathrm{d}\phi'\mathrm{d}z'$$
Then, using the property $\mathbf{J}^{(n)}(r',\phi',z') = a_n \mathbf{J}^{(0)}(r',\phi'-n\Delta\phi,z')$ ($a_n$ complex number) and taking advantage of the simple change of coordinates $\widetilde{\phi}=\phi'-n\Delta\phi$ we arrive at:
$$\mathcal{a}(\theta,\phi)=\sum_n a_n \int_{V_0} \mathbf{J}^{(0)}(r',\widetilde{\phi},z')e^{jk\left (r'\sin\theta\cos (\phi-\widetilde{\phi}-n\Delta\phi)+z'\cos\theta  \right )}r'\mathrm{d}r'\mathrm{d}\widetilde{\phi}\mathrm{d}z'$$
The latter formula, if calculated at observation points 'on the ground' ($\theta = \pi / 2 $, which is the hypothesis in which the book is put to come up with its formula (2.2) on page 20) is reduced to:
$$\mathcal{a}(\phi)=\sum_n a_n \underbrace{\int_{V_0} \mathbf{J}^{(0)}(r',\widetilde{\phi},z')e^{jk\left (r'\cos (\phi-\widetilde{\phi}-n\Delta\phi)  \right )}r'\mathrm{d}r'\mathrm{d}\widetilde{\phi}\mathrm{d}z'}_{\text{EL}(\phi-n\Delta\phi)}$$
in which the 'element factor' $\text {EL} (\phi-n \Delta \phi)$ has been written in compact form, that is the radiation integral of the single element of the array.
You can clearly see the difference with respect to formula (2.2) on the page cited above: a complex exponential factor is missing in the summation.
Where does it have to come from, and where am I wrong?
 A: The difference seems to be in the definition of element factor $EL_n$: according to equation (2.1) the amplitude of a single element is not $V_nEL_n(\mathbf{r}-\mathbf{R}_n)$, but
$$
E_{n}(\mathbf{r})=V_n EL_n(\mathbf{r}-\mathbf{R}_n)e^{-jk|\mathbf{r}-\mathbf{R}_n|}.
$$
Factor $e^{-jk|\mathbf{r}-\mathbf{R}_n|}$ is not included in $EL_n$.
UPDATE:
$EL_n$ and $V_n$ are not clearly defined in the book, that's why it is possible to get different results.
Consider an array of small elements (with size much less than the wavelength $\lambda$):
$$
a(\theta,\phi)=\sum_n\int_{V_n} \mathbf{J}_n(\mathbf{r'})e^{ik\mathbf{r}_0\mathbf{r}'}dV'=\sum_n\int_{V_0} \mathbf{J}_n(\mathbf{r'}+\mathbf{R}_n)e^{ik\mathbf{r}_0(\mathbf{r}'+\mathbf{R}_n)}dV'
$$
If we define $$
a_nEL_n(\theta,\phi)=\int_{V_0} \mathbf{J}_n(\mathbf{r'}+\mathbf{R}_n)e^{ik\mathbf{r}_0\mathbf{r}'}dV',
$$
then $a(\theta,\phi)$ can be written as:
$$
a(\theta,\phi)=\sum_n a_n EL_n(\theta,\phi) e^{ik\mathbf{r}_0\mathbf{R}_n}
$$
For a linear array, $EL_n(\theta,\phi)$ does not depend on $n$, which leads to formula (2.10). For a circular array, $EL_n(\theta,\phi)=EL_0(\theta,\phi-n\Delta\varphi)$, which leads to equation (2.11).
If $EL_n$ is defined in a different way, for example:
$$
a_n\tilde{EL}_n(\theta,\phi)=\int_{V_0} \mathbf{J}_n(\mathbf{r'}+\mathbf{R}_n)e^{ik\mathbf{r}_0(\mathbf{r}'+\mathbf{R}_n)}dV'
$$
then the result will be like yours: $a(\theta,\phi)=\sum_n a_n \tilde{EL}_n(\theta,\phi)$.
