Initial program 1.8
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
Initial simplification0.6
\[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot \left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{1 - z}\right) + \frac{-1259.1392167224028}{\left(1 - z\right) + 1}\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right)\right) + \left(\left(\frac{-0.13857109526572012}{7 - \left(z + 1\right)} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{1 - \left(z + -7\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) + 6}\right)\right)\right)\]
Taylor expanded around 0 1.4
\[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot \left(\left(\color{blue}{\left(361.7355639412844 \cdot z + \left(519.1279660315847 \cdot {z}^{2} + 47.95075976068347\right)\right)} + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right)\right) + \left(\left(\frac{-0.13857109526572012}{7 - \left(z + 1\right)} + \frac{12.507343278686905}{6 - \left(z + 1\right)}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{1 - \left(z + -7\right)} + \frac{9.984369578019572 \cdot 10^{-06}}{\left(1 - z\right) + 6}\right)\right)\right)\]
Taylor expanded around 0 1.4
\[\leadsto \left(\left(\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \frac{1}{e^{\left(1 - z\right) - \left(-6 - 0.5\right)}}\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot \left(\left(\left(361.7355639412844 \cdot z + \left(519.1279660315847 \cdot {z}^{2} + 47.95075976068347\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right)\right) + \color{blue}{\left(0.49644474017195733 \cdot z + \left(0.09941724278406094 \cdot {z}^{2} + 2.478374918352015\right)\right)}\right)\]
Taylor expanded around 0 1.3
\[\leadsto \left(\color{blue}{\left(\frac{1}{e^{7.5}} + \left(\frac{1}{2} \cdot \frac{z}{e^{7.5}} + \left(\frac{1}{6} \cdot \frac{z \cdot {\pi}^{2}}{e^{7.5}} + \frac{1}{z \cdot e^{7.5}}\right)\right)\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot \left(\left(\left(361.7355639412844 \cdot z + \left(519.1279660315847 \cdot {z}^{2} + 47.95075976068347\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right)\right) + \left(0.49644474017195733 \cdot z + \left(0.09941724278406094 \cdot {z}^{2} + 2.478374918352015\right)\right)\right)\]
Simplified1.3
\[\leadsto \left(\color{blue}{\left(\left(\frac{\frac{1}{z}}{e^{7.5}} + e^{-7.5}\right) + \left(\left(\pi \cdot \frac{1}{6}\right) \cdot \pi + \frac{1}{2}\right) \cdot \frac{z}{e^{7.5}}\right)} \cdot \left(\sqrt{2 \cdot \pi} \cdot {\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)}\right)\right) \cdot \left(\left(\left(361.7355639412844 \cdot z + \left(519.1279660315847 \cdot {z}^{2} + 47.95075976068347\right)\right) + \left(\frac{771.3234287776531}{1 - \left(z + -2\right)} + \frac{-176.6150291621406}{5 - \left(z + 1\right)}\right)\right) + \left(0.49644474017195733 \cdot z + \left(0.09941724278406094 \cdot {z}^{2} + 2.478374918352015\right)\right)\right)\]
Final simplification1.3
\[\leadsto \left(\left({\left(\left(1 - z\right) - \left(-6 - 0.5\right)\right)}^{\left(\left(1 - z\right) - \left(1 - 0.5\right)\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot \left(\left(\frac{\frac{1}{z}}{e^{7.5}} + e^{-7.5}\right) + \frac{z}{e^{7.5}} \cdot \left(\left(\pi \cdot \frac{1}{6}\right) \cdot \pi + \frac{1}{2}\right)\right)\right) \cdot \left(\left(\left(2.478374918352015 + 0.09941724278406094 \cdot {z}^{2}\right) + 0.49644474017195733 \cdot z\right) + \left(\left(\frac{-176.6150291621406}{5 - \left(z + 1\right)} + \frac{771.3234287776531}{1 - \left(-2 + z\right)}\right) + \left(\left({z}^{2} \cdot 519.1279660315847 + 47.95075976068347\right) + 361.7355639412844 \cdot z\right)\right)\right)\]
