Complex analysis applications - I

Calculus Level 5

1 2 ( 1 + i ) + ( 1 + i ) ∑ n = 1 ∞ 1 n 2 + ( 1 + i ) 2 \large \frac{1}{2(1 + i)} + (1 + i)\sum_{n = 1}^\infty \frac{1}{n^2 + (1 + i)^2}

Find the beautiful closed form for the sum above to 6 decimal places.

Notation: i = − 1 i = \sqrt{-1} denotes the imaginary unit .


The answer is 1.576674.

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

The solution is ,right now, here, Residue theorem applications: Series summation . The above sum has this closed form: It's is equal to π coth ⁡ π 2 ≈ 1.576674 \frac{\pi \coth \pi}{2} \approx 1.576674

( ( 1 + i ) ⋅ ∑ n = 1 ∞ 1 n 2 + ( 1 + i ) 2 ) + 1 2 ( 1 + i ) = π ⋅ ( 1 + e 2 π ( 1 + i ) ) 2 ⋅ ( e 2 π ( 1 + i ) − 1 ) = π coth ⁡ π 2 \displaystyle \left((1 + i)\cdot \sum_{n = 1}^\infty \frac{1}{n^2 + (1 + i)^2}\right) + \frac{1}{2(1 + i)} = \frac{\pi \cdot (1 + e^{2\pi (1 + i)})}{2\cdot (e^{2\pi (1 +i)} - 1)} = \frac{\pi \coth \pi}{2}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

This a nice wiki, I would look forward to contribute for applications of residues to special functions with examples soon !

Aditya Narayan Sharma - 4 years, 3 months ago

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You can contribute, I'm going to post another exercise about this, I'm going on to keep on with that wiki, thank you for your appreciation!!

Guillermo Templado - 4 years, 3 months ago

I have many ideas, Bernouilli numbers, Riemann Zeta function...

Guillermo Templado - 4 years, 3 months ago

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