<p dir="auto"><strong>Is your feature request related to a problem? Please describe.</strong></p>
<p dir="auto">Currently the incomplete elliptic integrals function in scipy.special do not support complex arguments for <code class="notranslate">phi</code></p>
<p dir="auto"><strong>Describe the solution you'd like</strong><br>
This could easily solved using the Landen Transformation. Let's take as example, the incomplete elliptic integral of the first kind <code class="notranslate">ellipkinc</code></p>
<p dir="auto">We could approximate the function as follows (note that it is expressed in terms of the parameter k instead of m where m=k²):</p>
<p dir="auto"><a target="_blank" rel="noopener noreferrer nofollow" href="https://user-images.githubusercontent.com/17838489/70849434-8851db80-1e7e-11ea-9465-d360f6cd4f60.png"><img src="https://user-images.githubusercontent.com/17838489/70849434-8851db80-1e7e-11ea-9465-d360f6cd4f60.png" alt="k_and_phi" style="max-width: 100%;"></a><br>
<a target="_blank" rel="noopener noreferrer nofollow" href="https://user-images.githubusercontent.com/17838489/70849445-bcc59780-1e7e-11ea-8b31-0b103d652d21.png"><img src="https://user-images.githubusercontent.com/17838489/70849445-bcc59780-1e7e-11ea-8b31-0b103d652d21.png" alt="expression" style="max-width: 100%;"></a></p>
<p dir="auto">from the second expression above,, we could express <code class="notranslate">phi</code> at <code class="notranslate">n+1</code> as follows:<br>
<a target="_blank" rel="noopener noreferrer nofollow" href="https://user-images.githubusercontent.com/17838489/70849510-a835cf00-1e7f-11ea-89fd-b7db105e1e16.png"><img src="https://user-images.githubusercontent.com/17838489/70849510-a835cf00-1e7f-11ea-89fd-b7db105e1e16.png" alt="phin1" style="max-width: 100%;"></a></p>
<p dir="auto">The advantage of this last expression over using something like <code class="notranslate">scipy.optimize.fsolve</code> is that it could accept complex numbers.</p>
<p dir="auto">The implementation is easy.</p>
<p dir="auto">For different complex arguments for <code class="notranslate">phi</code> the results match exactly with results in WolframAlpha</p>
<p dir="auto"><strong>Reference</strong></p>
<blockquote>
<p dir="auto">R.E. Deakin &amp; M.N. Hunter Elliptic Integrals And Landen's Transformation: What I should've known  for Geodesy. <a href="http://www.mygeodesy.id.au/documents/Elliptic%20Integrals%20and%20Landen%27s%20Transformation.pdf" rel="nofollow">Link</a></p>
</blockquote>
<p dir="auto">Continuation of <a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="13650440" data-permission-text="Title is private" data-url="https://github.com/scipy/scipy/issues/1431" data-hovercard-type="issue" data-hovercard-url="/scipy/scipy/issues/1431/hovercard" href="https://github.com/scipy/scipy/issues/1431">gh-1431</a>: extend <code class="notranslate">ellipk</code> et al to complex plane.</p>