Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
Piecewise((-i*x^2*acosh(y/x)/2+i*y^3/(2*x*sqrt(-1+y^2/x^2))-i*x*y/(2*sqrt(-1+y^2/x^2)),|y^2/x^2|>1),(x^2*asin(y/x)/2+x*y*sqrt(1-y^2/x^2)/2,True))
¿Cómo vas a descomponer esta Piecewise((-i*x^2*acosh(y/x)/2+i*y^3/(2*x*sqrt(-1+y^2/x^2))-i*x*y/(2*sqrt(-1+y^2/x^2)),|y^2/x^2|>1),(x^2*asin(y/x)/2+x*y*sqrt(1-y^2/x^2)/2,True)) expresión en fracciones?
Solución
Ha introducido
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/ 2 /y\ | I*x *acosh|-| 3 | 2| | \x/ I*y I*x*y |y | |- ------------- + ------------------- - ----------------- for |--| > 1 | 2 _________ _________ | 2| | / 2 / 2 |x | | / y / y | 2*x* / -1 + -- 2* / -1 + -- | / 2 / 2 < \/ x \/ x | | ________ | / 2 | / y | 2 /y\ x*y* / 1 - -- | x *asin|-| / 2 | \x/ \/ x | ---------- + ------------------ otherwise \ 2 2
$$\begin{cases} - \frac{i x^{2} \operatorname{acosh}{\left(\frac{y}{x} \right)}}{2} - \frac{i x y}{2 \sqrt{-1 + \frac{y^{2}}{x^{2}}}} + \frac{i y^{3}}{2 x \sqrt{-1 + \frac{y^{2}}{x^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{x^{2}}}\right| > 1 \\\frac{x^{2} \operatorname{asin}{\left(\frac{y}{x} \right)}}{2} + \frac{x y \sqrt{1 - \frac{y^{2}}{x^{2}}}}{2} & \text{otherwise} \end{cases}$$
Piecewise((-i*x^2*acosh(y/x)/2 + i*y^3/(2*x*sqrt(-1 + y^2/x^2)) - i*x*y/(2*sqrt(-1 + y^2/x^2)), |y^2/x^2| > 1), (x^2*asin(y/x)/2 + x*y*sqrt(1 - y^2/x^2)/2, True))
Simplificación general
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/ / _________ \ | | / 2 2 | | | / y - x /y\| |I*x*|y* / ------- - x*acosh|-|| | | / 2 \x/| | 2| | \ \/ x / |y | |------------------------------------ for |--| > 1 | 2 | 2| | |x | < | / _________\ | | / 2 2 | | | /y\ / x - y | | x*|x*asin|-| + y* / ------- | | | \x/ / 2 | | \ \/ x / | --------------------------------- otherwise | 2 \
$$\begin{cases} \frac{i x \left(- x \operatorname{acosh}{\left(\frac{y}{x} \right)} + y \sqrt{\frac{- x^{2} + y^{2}}{x^{2}}}\right)}{2} & \text{for}\: \left|{\frac{y^{2}}{x^{2}}}\right| > 1 \\\frac{x \left(x \operatorname{asin}{\left(\frac{y}{x} \right)} + y \sqrt{\frac{x^{2} - y^{2}}{x^{2}}}\right)}{2} & \text{otherwise} \end{cases}$$
Piecewise((i*x*(y*sqrt((y^2 - x^2)/x^2) - x*acosh(y/x))/2, |y^2/x^2| > 1), (x*(x*asin(y/x) + y*sqrt((x^2 - y^2)/x^2))/2, True))
Respuesta numérica
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Piecewise((-0.5*i*x^2*acosh(y/x) + 0.5*i*y^3*(-1.0 + y^2/x^2)^(-0.5)/x - 0.5*i*x*y*(-1.0 + y^2/x^2)^(-0.5), |y^2/x^2| > 1), (0.5*x^2*asin(y/x) + 0.5*x*y*(1.0 - y^2/x^2)^0.5, True))
Piecewise((-0.5*i*x^2*acosh(y/x) + 0.5*i*y^3*(-1.0 + y^2/x^2)^(-0.5)/x - 0.5*i*x*y*(-1.0 + y^2/x^2)^(-0.5), |y^2/x^2| > 1), (0.5*x^2*asin(y/x) + 0.5*x*y*(1.0 - y^2/x^2)^0.5, True))
Unión de expresiones racionales
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/ / _________ \ | | / 2 2 | | | 3 2 3 / y - x /y\| |I*|y - y*x - x * / ------- *acosh|-|| | | / 2 \x/| | 2| | \ \/ x / |y | |------------------------------------------- for |--| > 1 | _________ | 2| | / 2 2 |x | | / y - x | 2*x* / ------- < / 2 | \/ x | | / _________\ | | / 2 2 | | | /y\ / x - y | | x*|x*asin|-| + y* / ------- | | | \x/ / 2 | | \ \/ x / | --------------------------------- otherwise | 2 \
$$\begin{cases} \frac{i \left(- x^{3} \sqrt{\frac{- x^{2} + y^{2}}{x^{2}}} \operatorname{acosh}{\left(\frac{y}{x} \right)} - x^{2} y + y^{3}\right)}{2 x \sqrt{\frac{- x^{2} + y^{2}}{x^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{x^{2}}}\right| > 1 \\\frac{x \left(x \operatorname{asin}{\left(\frac{y}{x} \right)} + y \sqrt{\frac{x^{2} - y^{2}}{x^{2}}}\right)}{2} & \text{otherwise} \end{cases}$$
Piecewise((i*(y^3 - y*x^2 - x^3*sqrt((y^2 - x^2)/x^2)*acosh(y/x))/(2*x*sqrt((y^2 - x^2)/x^2)), |y^2/x^2| > 1), (x*(x*asin(y/x) + y*sqrt((x^2 - y^2)/x^2))/2, True))