test whether the function x²(x-iy)/x⁴+y² is analytic and continuous

Question

complex analysis

Answer 1

Assuming x and y belong to R, then the expression can be split into x^3/(x^4+y^4) and -ix^2y/(x^4+y^4). When x and y are both zero the complex expression appears to be undefined because denominator and numerator both become zero in both components. This makes the complex expression indefinable and therefore non-continuous. The exponents in the denominator are of higher degree than the numerator and so effectively make the expression infinite when x and y are zero. The discontinuity makes them non-analytic because of this singularity (the components cannot be differentiated).

I think!