horizontal alignment of equations across multiple lines

Question

I need the following to line up so that the first line (integral) is on one line on the left at the top, then the following lines are indented and aligned to each other, but also so the 2 sets of conditions (lines starting \left[\alpha>0,... and \left[\Real\mu...) will align to be flush with the right of the page. How do I do this?

Thanks

\begin{document}

\begin{multline}
\int_0^\infty e^{-\alpha x}\, Y_\nu \left(\beta x\right)x^{\mu - 1}\,dx\\
= \cot v\pi\frac{\left(\frac{\beta}{2}\right)^{\nu}\Gamma\left(\nu+\mu\right)}{\sqrt{\left(\alpha^{2}+\beta^{2}\right)^{\nu+\mu}}\Gamma\left(\nu+1\right)}F\left(\frac{\nu+\mu}{2},\frac{\nu-\mu+1}{2};\nu+1;\frac{\beta^{2}}{\alpha^{2}+\beta^{2}}\right)\\
= \mathrm{cosec}\, \nu\pi\frac{\left(\frac{\beta}{2}\right)^{-\nu}\Gamma\left(\mu-v\right)}{\sqrt{\left(\alpha ^{2}+\beta^{2}\right)^{\mu-\nu}}\Gamma\left(1-\nu\right)}F\left(\frac{\mu-\nu}{2},\frac{1-\nu-\mu}{2};1-\nu;\frac{\beta^{2}}{\alpha^{2}+\beta^{2}}\right)\\
\left[\Real \mu \geq\lvert \Real \nu\rvert,\quad\Real\left(\alpha\pm \textit{i}\beta\right)>0\right]\\
=-\frac{2}{\pi}\Gamma\left(\nu+\mu\right)\left(\beta^{2}+\alpha^{2}\right)^{-\frac{1}{2}\mu} Q_{\mu-1}^{-\nu}\left[\alpha\left(\alpha^{2}+\beta^{2}\right)^ {-\frac{1}{2}}\right]\\
\left[\alpha>0,\quad\beta>0,\quad\Real\mu>\lvert\Real \nu\rvert\right]
\end{multline}

\end{document}

Answer 1

Like this?

I suppose that \Real means \Re ... The MWE for above equations is

\documentclass{article}
\usepackage{mathtools}
\DeclareMathOperator{\cosec}{cosec}

\begin{document}

\begin{align}
    \MoveEqLeft
\int_0^\infty e^{-\alpha x}\, Y_\nu \left(\beta x\right)x^{\mu - 1}\,dx
                \notag  \\
    & = \cot v\pi\frac{\left(\frac{\beta}{2}\right)^{\nu}
        \Gamma\left(\nu+\mu\right)}{\sqrt{\left(\alpha^{2}+\beta^{2}\right)^{\nu+\mu}}
        \Gamma\left(\nu+1\right)}
    F\left(\frac{\nu+\mu}{2},\frac{\nu-\mu+1}{2};\nu+1;\frac{\beta^{2}}{\alpha^{2}+\beta^{2}}\right)
                \notag  \\
    & = \cosec\, \nu\pi\frac{\left(\frac{\beta}{2}\right)^{-\nu}
        \Gamma\left(\mu-v\right)}{\sqrt{\left(\alpha ^{2}+\beta^{2}\right)^{\mu-\nu}}\Gamma\left(1-\nu\right)}
        F\left(\frac{\mu-\nu}{2},\frac{1-\nu-\mu}{2};1-\nu;\frac{\beta^{2}}{\alpha^{2}+\beta^{2}}\right)
                \notag  \\
\intertext{\raggedleft    
    $\left[\Re \mu \geq\lvert \Re \nu\rvert,
     \quad\Re\left(\alpha\pm \textit{i}\beta\right)>0\right]
    $}
    & =-\frac{2}{\pi}\Gamma\left(\nu+\mu\right)\left(\beta^{2}+\alpha^{2}\right)^{-\frac{1}{2}\mu} 
        Q_{\mu-1}^{-\nu}\left[\alpha\left(\alpha^{2}+\beta^{2}\right)^ {-\frac{1}{2}}\right]
\intertext{\raggedleft
    $\left[\alpha>0,\quad\beta>0,\quad\Re \mu>\lvert\Re \nu\rvert\right]
    $}
                \notag  
\end{align}

\end{document}

Answer 2

You may want to use a combination of equation and split environments. (I wouldn't shoving the two condition lines all the way to the right.)

\documentclass{article}
\usepackage{amsmath,amsfonts}
\DeclareMathOperator{\cosec}{cosec}
\newcommand\Real{\mathfrak{R}} % is this correct?!
\usepackage{mleftright}\mleftright
\begin{document}

\begin{equation}\begin{split}
&\int_0^\infty e^{-\alpha x}\, Y_\nu (x)x^{\mu - 1}\,dx\\
&= \cot \nu\pi\frac{\bigl(\frac{\beta}{2}\bigr)^{\nu}\,\Gamma(\nu+\mu)}{\sqrt{(\alpha^2+\beta^2{)}^{\nu+\mu}}\,\Gamma(\nu+1)}
F\left(\frac{\nu+\mu}{2},\frac{\nu-\mu+1}{2};\nu+1;\frac{\beta^2}{\alpha^2+\beta^2}\right)\\
&= \cosec \nu\pi\frac{\bigl(\frac{\beta}{2}\bigr)^{-\nu}\,\Gamma(\mu-v)}{\sqrt{(\alpha^2+\beta^2{)}^{\mu-\nu}}\,\Gamma(1-\nu)}
F\left(\frac{\mu-\nu}{2},\frac{1-\nu-\mu}{2};1-\nu;\frac{\beta^2}{\alpha^2+\beta^2}\right)\\[1ex]
&\qquad\qquad\bigl[\Real\mu\geq\lvert \Real \nu\rvert,\ 
 \Real(\alpha\pm i\beta)>0\bigr]\\[1ex]
&=-\frac{2}{\pi} \Gamma(\nu+\mu)(\beta^2+\alpha^2{)}^{-(1/2)\mu}\, Q_{\mu-1}^{-\nu}
\bigl[\alpha(\alpha^2+\beta^2{)}^{-1/2}\,\bigr]\\[1ex]
&\qquad\qquad\bigl[\alpha>0,\ \beta>0,\ \Real\mu>\lvert\Real \nu\rvert\bigr]
\end{split}\end{equation}

\end{document}