Question about a theorem of Littlewood I'm reading the book "The Theory of Functions" by E.C. Titchmarsh. He shows the proof of a theorem of Littlewood (Titchmarsh refers to: J.E. Littlewood, "On the zeros of the Riemann zeta-function", Proc. Camb. Phil. Soc. 22 (1924), 295-318). In its simplest form, the theorem states that the contour integral of $log (z-w)$ around a rectangle with sides $x_1$, $x_2$, $y_1$, $y_2$ (with $x_1<x_2$ and $y_1<y_2$), where $w=\alpha+i\beta$ lies within the rectangle, equals $2\pi i(x_1-\alpha)$. What I don't understand is why the result does not depend on $x_2$, $y_1$, $y_2$ or $\beta$. Could someone explain that to me, please?
 A: The Littlewood theorem you are referring to is a variant of Cauchy's integral theorem, which states that the contour integral of a holomorphic function around a closed path is zero, as long as the function is holomorphic in the region enclosed by the path.
In this case, we are taking the contour integral of $log(z-w)$ around a rectangle with sides $x_1$, $x_2$, $y_1$, $y_2$. The function $log(z-w)$ is holomorphic in the region enclosed by the rectangle, as long as $w=\alpha+i\beta$ lies within the rectangle.
The complex logarithm function is defined as $log(z) = log|z| + iArg(z)$, where $Arg(z)$ is the argument of $z$, which is the angle between the positive real axis and the vector pointing to $z$. It also important to note that in the case of the Littlewood theorem, the logarithm is not considered as a multi-valued function, it's a single-valued function. The integration is done around a rectangle and the logarithm function is defined on this rectangle. It's defined as $log(z-w) = log(z-\alpha-i\beta) = log(|z-\alpha-i\beta|)+iArg(z-\alpha-i\beta)$ where $\alpha$ and $\beta$ are fixed numbers, so it's not a multi-valued function anymore, it's single-valued.
When we integrate this function over the rectangle, we are summing over all possible values of the function, which means that the integral is independent of the specific values of $x_2$, $y_1$, $y_2$, and $\beta$, because they only affect the value of the argument of $z$, and the integral is over the whole range of the argument.
So, the result only depends on the value of $\alpha$ which is the real part of the point $w$ and the value of $x_1$ which is the lower limit of the contour integral, and the integral of the logarithm function around the rectangle with these value is $2\pi i (x_1-\alpha)$.
