#26 Measure Your Calibre

$\begin{aligned} \displaystyle\int_{0}^{1} \left[ \dfrac{256 \cdot i^{6} \cdot \dfrac{d^{2}}{d \theta^{2}} \left \{ \displaystyle\prod_{r=1}^{503} e^{(2r-1)i \theta} \right \} }{ \displaystyle\prod_{r=0}^{502} e^{(2r+1)i \theta} } \right]^{{1}/{4}} d\theta &= \displaystyle\int_{0}^{1} \left[ \dfrac{256 \cdot i^{6} \cdot \dfrac{d^{2}}{d \theta^{2}} \left \{ e^{i \theta (503)^2} \right \} }{ e^{i \theta (503)^2} } \right]^{{1}/{4}} d\theta & \small {\color{#3D99F6} (*)}\\ &= \displaystyle\int_{0}^{1} \left[ \dfrac{256 \cdot i^{6} \cdot \left \{ e^{i \theta (503)^2} \cdot i^2 \cdot (503)^4 \right \} }{ e^{i \theta (503)^2} } \right]^{{1}/{4}} d\theta & \small {\color{#3D99F6} (**)}\\ &= \displaystyle\int_{0}^{1} \left[ 256 \cdot (503)^4 \underbrace{i^8}_{\color{#3D99F6} = \ 1} \right]^{{1}/{4}} d\theta \\ &= 4 \cdot 503 \displaystyle\int_{0}^{1} d\theta \\ &= \boxed{2012} \end{aligned}$

$\begin{aligned} \large \color{#3D99F6} (*) \ \ & \displaystyle\prod_{r=1}^{503} e^{(2r-1)i \theta} = \exp \left\{ \displaystyle \sum_{r=1}^{503} (2r-1) i \theta \right\} = \exp \left\{i \theta \cdot (503)^2 \right\} \\ & \displaystyle\prod_{r=0}^{502} e^{(2r+1)i \theta} = \displaystyle\prod_{r=1}^{503} e^{(2r-1)i \theta} = \exp \left\{ \displaystyle \sum_{r=1}^{503} (2r-1) i \theta \right\} = \exp \left\{i \theta \cdot (503)^2 \right\} \end{aligned}$

${\large \color{#3D99F6} (**)} \ \ \dfrac{d^{2}}{d \theta^{2}} e^{i \theta (503)^2} = i (503)^2 \cdot \dfrac{d}{d \theta} e^{i \theta (503)^2} = i^2 (503)^4 \cdot e^{i \theta (503)^2}$