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Complex Partial Fractions

Age 16 to 18 Challenge Level:

Find real constants $A, B$ and $C$ and complex constants $D$ and $E$ such that $${10x^2-2x+4\over x^3 + x} = {A\over x} +{Bx+C\over x^2+1} = {A\over x} + {D\over x-i} + {E\over x+i}.$$

NOTES AND BACKGROUND

This problem gives an example where a rational function can be reduced to a sum of linear partial fractions IF we allow ourselves to use complex numbers. It turns out that this is always possible! This is of use in more advanced university-level applications of integration and analysis of series.