How to have two successive but different alignments?

Question

in written codes 2 equality and 2 pluses are aligned. What I want is to align 2 equality in one hand, and 3 limits on the other hand. I thought that defining another "split" before the second limit would work, but it did not. Would you mind helping me out?

\begin{equation}
\begin{split}
w(z)
&=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!} z^{n}\int_{\mathcal{C}}t^{(\beta+n)-1}(1-t)^{(\gamma-\beta)-1}dt\\
&=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!} z^{n}[\lim_{\sigma\rightarrow0}\int_{0}^{1-\sigma}t^{(\beta+n)-1}(1-t)^{(\gamma-\beta)-1}dt\\
&+\underbrace{\lim_{\sigma\rightarrow0}\int_{circle}t'^{(\beta+n)-1}(1-t')^{(\gamma-\beta)-1}dt'}_{0}\\
&+\lim_{\sigma\rightarrow0}\int_{1-\sigma}^{0}t''^{(\beta+n)-1}(1-t'')^{(\gamma-\beta)-1}dt'']
\end{split}
\end{equation}

Answer 1

Nest an aligned[t] environment within split. Unrelated: the index _{circle} is typed as the product of the six variables c, i, r, c, l, e, with the spacing of variables. Use \text, or \mathrm.

\documentclass[11pt]{article}
\usepackage{mathtools, nccmath}

\begin{document}

 \begin{equation}
\begin{split}
w(z)
&=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!} z^{n}\int_{\mathcal{C}}t^{(\beta+n)-1}(1-t)^{(\gamma-\beta)-1}dt\\
&=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!} z^{n}
\biggl[\begin{aligned}[t] & \lim_{\sigma\rightarrow0}\int_{0}^{1-\sigma}t^{(\beta+n)-1}(1-t)^{(\gamma-\beta)-1}dt\\
 & +\underbrace{\lim_{\sigma\rightarrow0}\int_{\text{circle}}t'^{(\beta+n)-1}(1-t')^{(\gamma-\beta)-1}dt'}_{0}\\[-0.5ex]
 & + \lim_{\sigma\rightarrow0}\int_{1-\sigma}^{0}t''^{(\beta+n)-1}(1-t'')^{(\gamma-\beta)-1}dt''\biggr]
\end{aligned}
\end{split}
\end{equation}

\end{document}