I'm having a hard time to understand how's Eq. $(6.73)$ become Eq. $(6.75)$. It's taken from Numerical Methods for Laplace Transform Inversion by Cohen. Here's the problem:
[...]. The basis of their formulation is, like Talbot, that the inverse transform is given by $$f(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}e^{st}\bar f(s)ds.\tag{6.71}$$ The contour is deformed by means of the path $$s(\theta)=r\theta(\cot\theta+i),\quad-\pi<\theta<\pi,\tag{6.72}$$ where $r$ is a parameter. This path only involves one parameter whereas Talbot's consisted of two. Integration over the deformed path yields $$f(t)=\frac{1}{2\pi i}\int_{-\pi}^\pi e^{ts(\theta)}\bar f(s(\theta))s'(\theta)d\theta.\tag{6.73}$$ Differentiating (6.72) we have $s'(\theta)=ir(1+i\sigma(\theta))$, where $$\sigma(\theta)=\theta+(\theta\cot\theta-1)\cot\theta.\tag{6.74}$$ We find $$f(t)=\frac{r}{\pi}\int_0^\pi\Re\Bigl[e^{ts(\theta)}\bar f(s(\theta))(1+i\sigma(\theta))\Bigr]d\theta.\tag{6.75}$$
As you can see there, i'm confused about where the $\Re$ got from. And i have 2 speculations about this.
Speculation 1 (The idea is originated from my friend, so let me rewrite what he said)
Regarding the integral: $s(\theta)$ on $(-\pi,\,\pi)$ is symmetric about $\theta=0$ (i don't understand this part). So we can integrate on $\theta\in (0,\pi)$ and multiply by $2$ in the denominator of the constant (outside the integral). The factor $ir$ in $s'(\theta)$ cancels the $i$ in the constant and gives it $r$ in the numerator.
Assuming $f(t)$ is a real-valued function, $f(t)=\Re f(t)$. For Riemann integrable functions $g(z)$, we have $\Re \int g(z)\Bbb dz = \int \Re g(z) \Bbb dz$. Hence, since $f(t)$ is equal to an integral, we can take $\Re$ on both sides and move the real part symbol inside the integral.
Speculation 2 (This is my own idea)
By letting the whole integrand with $\Lambda$ for simplification purpose (i'm not performing change of variable here, just for a simplification). Then we can consider:
$$\begin{align} \int_{-\pi}^{\pi} \Lambda\,\Bbb d\Lambda &= \int_{-\pi}^{0} \Lambda\,\Bbb d\Lambda + \int_{0}^{\pi} \Lambda\,\Bbb d\Lambda\\ &= -\int_{0}^{-\pi} \Lambda\,\Bbb d\Lambda + \int_{0}^{\pi} \Lambda\,\Bbb d\Lambda\\ &= \int_{0}^{\pi} \overline{\Lambda}\,\Bbb d\overline{\Lambda} + \int_{0}^{\pi} \Lambda\,\Bbb d\Lambda\\ &= 2\int_{0}^{\pi}\Re \Lambda\,\Bbb d \Lambda \end{align}$$
The last expression is considering this property : $z + \overline{z} = 2\Re z$, where $\overline{z}$ is the conjugate of $z$. But i'm not really sure if i can consider the third line is valid about the conjugate? And by the way. The first equation on the picture is the formula for finding the Inverse Laplace Transform of $f(t)$ (real-valued) and the Laplace transform that what i'm talking is one-sided Laplace Transform.
Are both speculations correct? Is my speculation is correct? If not, please provide the correct interpretation behind the story of Eq. $(6.73)$ become Eq. $(6.75)$. Please kindly to help me, i really need your help. Thanks in advance!