Solve √(δx²+δy²)
You asked:
MathBot Answer:
Evaluated
\(\displaystyle \sqrt{δ \cdot {x}^{2} + δ \cdot {y}^{2}} = \sqrt{x^{2} δ + y^{2} δ} \)
Expanded
\[\sqrt{δ \cdot {x}^{2} + δ \cdot {y}^{2}} = \sqrt{x^{2} δ + y^{2} δ}\]
Factored
\[\sqrt{δ \cdot {x}^{2} + δ \cdot {y}^{2}} = \sqrt{δ \left(x^{2} + y^{2}\right)}\]
Roots
\[\sqrt{δ \cdot {x}^{2} + δ \cdot {y}^{2}} = i \sqrt[4]{\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)} \approx i \left(\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)} + \left(\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)}\]\[\sqrt{δ \cdot {x}^{2} + δ \cdot {y}^{2}} = i \left(- \sqrt[4]{\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)}\right) - \sqrt[4]{\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)} \approx - i \left(\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)} - \left(\left(\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)}\right)^{2} + \left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x^{2} δ\right)} + \operatorname{im}{\left(y^{2} δ\right)},\operatorname{re}{\left(x^{2} δ\right)} + \operatorname{re}{\left(y^{2} δ\right)} \right)}}{2} \right)}\]