I am having trouble simplifying an equation that describes expected momentum of a wave function. The problem asks us to express expected momentum $〈\space p \space 〉$ in simplified terms.
$$(1)\space\space\space\space\space\space\space\space\space\space\space\space〈\space p \space 〉 =\int_{-\infty}^\infty \phi\dot(x)\space \hbar\space (-i)\frac{\delta}{\delta x}\phi(x)\space\delta x$$
where $\phi\dot(x)$ is the complex conjugate of $\phi(x)$
By definition of the fourier transform
$$(2)\space\space\space\space\space\space\space\space\space\space\space\space\phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \hat \phi(k) \space e^{ikx} \delta k $$
By substituting the fourier transform in for each $\phi(x)$ and $\phi\dot(x)$ we obtain
$$(3)\space\space\space\space\space\space\space\space\space\space\space\space〈\space p \space 〉 =\int_{-\infty}^\infty \frac{1}{2\pi}\Bigg(\int_{-\infty}^{\infty} \hat \phi(j) \space e^{ijx} \delta j\Bigg)\space\space \hbar\space (-i)\frac{\delta}{\delta x}\Bigg(\int_{-\infty}^{\infty} \hat \phi(k) \space e^{ikx} \delta k\Bigg)\delta x$$
This is correct according to the solution guide. I don't understand why the complex conjugate form of $\hat\phi\dot(j)$ is not preserved. Instead the solution shows $\hat\phi(j)$.
The problem goes on to simplify further $$(4)\space\space\space\space\space\space\space\space\space\space\space\space〈\space p \space 〉 =\int_{-\infty}^\infty \frac{1}{2\pi}\Bigg(\int_{-\infty}^{\infty} \hat \phi\dot(j) \space e^{-ijx} \delta j\Bigg)\space\space \Bigg(\int_{-\infty}^{\infty} \frac{\delta}{\delta x} \hbar\space (-i)\space \hat \phi(k) \space e^{ikx} \delta k\Bigg)\delta x$$
Equation 4 brings in $\frac\delta{\delta x} i\hbar$ within the integral and brings back the complex conjugate form $\hat \phi \dot(j)$. Notice there is a negative coefficient on the superscript of $e^{-ijx}$
There must be a principle of mathematics that I am unaware of. What happened to the complex conjugate form of $\phi\dot(x)$ and what brings it back after making the coefficient of the superscript negative?
Most likely, it is a typing mistake. Take the complex conjugate of the LHS and RHS of (2) to get $\dot{\phi(x)}$. Equation (4) shows the correct integral representaiton. But I am not sure why $\sqrt{2\pi}$ instead of $2\pi$ is appearing in (3) and (4).