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certificate-n-1-n-4-a-4-2-a-3-sinh-2-a-sin-2-a-2-a-3-cosh-2-a-cos-2-a-




Question Number 211566 by MrGaster last updated on 13/Sep/24
                 certificate:          Σ_(n=−∞) ^(+∞) (1/(n^4 +a^4 ))=(𝛑/( (√2)a^3 )) ((sinh((√2)a𝛑)+sin((√2)a𝛑))/( (√2)a^3 cosh((√2)a𝛑)−cos((√2)a𝛑)))
$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\underset{\boldsymbol{{n}}=−\infty} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{4}} +\boldsymbol{{a}}^{\mathrm{4}} }=\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} }\:\frac{\boldsymbol{\mathrm{si}{n}\mathrm{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)+\boldsymbol{\mathrm{sin}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} \boldsymbol{\mathrm{cos}{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)−\boldsymbol{\mathrm{cos}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)} \\ $$$$ \\ $$

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