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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC