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Fungrim entry: 6ddbf4

abzdz=23(b3/2a3/2)\int_{a}^{b} \sqrt{z} \, dz = \frac{2}{3} \left({b}^{3 / 2} - {a}^{3 / 2}\right)
Assumptions:aCandbCand((Re(a)0andRe(b)0)or(Im(a)0andIm(b)0)or(Im(a)<0andIm(b)<0))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:aCandbCand(a,b)(,0)={}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
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ImIm(z)\operatorname{Im}(z) Imaginary part
OpenInterval(a,b)\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()))))

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2020-01-31 18:09:28.494564 UTC