I am trying understand derivation of fourier series from $2\pi$-periodic funtions to $T$-periodic functions. Here, this is for $2\pi$-periodic functions
$$\frac{1}{\sqrt{2\pi}}\int _{-\pi}^{\pi} x(t^*)\,e^{-jnt^*} dt^* = X(n)$$ $$x(t^*)=\frac{1}{\sqrt{2\pi}}\sum _{n=-\infty}^{\infty}X(n)\,e^{jnt^*}$$
Which is I totally understood reasoning behind it. I converted them by using following relation. $$\frac{t^*}{2\pi} = \frac{t}{T}$$ Finally I got this: $$x(t) = \frac{1}{\sqrt{2\pi}} \sum_{n=-\infty}^{\infty} X(n) \, e^{j\frac{2\pi}{T}nt}$$ $$X(n) = \frac{\sqrt{2\pi}}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} x(t) \, e^{-jn\frac{2\pi}{T} t} \, dt$$
However, in my class notes, it was written without $\sqrt{2\pi}$ coefficient, so it is like transforming ${\sqrt{2\pi}} x(t)$ to $X(n)$ instead of $x(t)$ to $X(n)$. My question is why there is such difference? And does not it cause an error?
Edit: My class notes: $$x(t) = \sum_{n=-\infty}^{\infty} X(n) \, e^{j\frac{2\pi}{T} nt}$$ $$X(n) = \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} x(t) \, e^{-jn\frac{2\pi}{T} t} \, dt$$