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Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
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How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?
The answer is not unique - you can add any function of period $a$ to f. This is what the singularities are trying to tell you. In a distributional sense, the Fourier transform of a function of period …