Consider this simple equation $$ \tau(t)=\int\frac{dt}{a(t)},$$ where $\tau$ and $a$ are functions of $t$. Now, from this equation, how can I calculate $a(\tau)$ ?

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    $\begingroup$ You should not use the same symbol for the limits of integration and the variable of integration. $\endgroup$ – Kavi Rama Murthy Mar 18 at 6:21
  • $\begingroup$ Thanks a lot. I have deleted limits in my question. @Kavi Rama Murthy $\endgroup$ – Photon Mar 18 at 6:24

$\tau'(t)=\frac 1 {a(t)}$ so $a(t)=\frac 1 {\tau'(t)}$.

  • $\begingroup$ Thanks a lot. But I need an equation of a as a function of $\tau$ i.e., the equation of $a(\tau$) not $a(t)$. @Kavi Rama Murthy $\endgroup$ – Photon Mar 18 at 6:27
  • $\begingroup$ $\tau$ is a function of $t$, not a number. If you want to find the function $a\circ \tau$ the answer is $a(\tau (t))=1/\tau'(\tau (t))$. $\endgroup$ – Kavi Rama Murthy Mar 18 at 6:31

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