Fungrim entry: 6ddbf4

\int_{a}^{b} \sqrt{z} \, dz = \frac{2}{3} \left({b}^{3 / 2} - {a}^{3 / 2}\right)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(\operatorname{Re}(a) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(a) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(a) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) < 0\right)\right)

Alternative assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a, b\right) \cap \left(-\infty, 0\right) = \left\{\right\}

TeX:

\int_{a}^{b} \sqrt{z} \, dz = \frac{2}{3} \left({b}^{3 / 2} - {a}^{3 / 2}\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(\operatorname{Re}(a) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(b) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(a) \ge 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(\operatorname{Im}(a) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) < 0\right)\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(a, b\right) \cap \left(-\infty, 0\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`Im`	$\operatorname{Im}(z)$	Imaginary part
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("6ddbf4"),
    Formula(Equal(Integral(Sqrt(z), For(z, a, b)), Mul(Div(2, 3), Sub(Pow(b, Div(3, 2)), Pow(a, Div(3, 2)))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Or(And(GreaterEqual(Re(a), 0), GreaterEqual(Re(b), 0)), And(GreaterEqual(Im(a), 0), GreaterEqual(Im(b), 0)), And(Less(Im(a), 0), Less(Im(b), 0)))), And(Element(a, CC), Element(b, CC), Equal(Intersection(OpenInterval(a, b), OpenInterval(Neg(Infinity), 0)), Set()))))

Topics using this entry

Square roots