View source for Square root (real function)

{{Stub page|grade=C|msg=Made this page to prove smoothness really [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 05:46, 10 April 2017 (UTC)}}


: '''Note: ''' For other uses of square root see [[Square root|''square root (disambiguation)'']] - this page only covers the square root as a function on the non-negative reals.
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==Definition==
Let {{M|x\in\mathbb{R}_{\ge 0} }} be given, then to say {{M|y}} is a ''square root'' of {{M|x}} means that {{M|y^2\eq x}}, note that if {{M|y^2\eq x}} then {{M|-y}} is also a square root of {{M|x}}:
* Notice: {{M|(-y)^2\eq(-1)^2 y^2\eq (-1)^2 x}} and that {{M|-1\times -1\eq 1}}, so {{M|(-y)^2\eq x}} also.
Note also however:
* If {{M|a^2\eq 0}} then we must have {{M|a\eq 0}}, for if {{M|a\neq 0}} then {{M|aa\neq 0}} - contradicting that {{M|a}} is a square-root. 
* As such {{M|0}} has only one square root, {{M|0}} itself.


Given {{M|x\in\mathbb{R}_{\ge 0} }} we write:
* {{M|\sqrt{x}\in\mathbb{R}_{\ge 0} }} - called the ''principle square root'' - a number such that {{M|\sqrt{x}\times\sqrt{x}\eq x}} and such that {{M|\sqrt{x}\in\mathbb{R} }} and {{M|\sqrt{x}\ge 0}}
* {{M|-\sqrt{x}\in\mathbb{R}_{\le 0} }} - called the ''negative square root'' - this is just {{M|(-1)\sqrt{x} }}
* {{M|\pm\sqrt{x} }} to emphasise there are possibly two of them, if there is one this becomes {{M|\pm 0}}, and {{M|x\pm 0\eq x}} so it doesn't matter.
==Evaluating the square root==
{{XXX|Flesh this out}}

* {{M|\sqrt{x}\eq e^{\frac{1}{2}\text{ln}(x)} }} where {{M|e}} is [[Euler's number]] and {{M|\text{ln}(x)}} is the [[natural logarithm]] of {{M|x}}
==Properties of the square-root function==
Let {{M|f:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R} }} be a [[function]] given by {{M|f:x\mapsto \sqrt{x} }}, then we claim:
* {{M|f\big\vert_{\mathbb{R}_{>0} }:\mathbb{R}_{>0}\rightarrow\mathbb{R} }} is [[smooth function|smooth]]
{{Requires proof|grade=B|Presley's Elementary Differential Geometry claims it can be done easily by induction on page 15. It claims:
* {{MM|\frac{\mathrm{d}^n f}{\mathrm{d}x^n}\eq(-1)^{n-1}\cdot\frac{1\times 3\times 5\times\cdots\times(2n-1)}{2^n}\cdot x^{-\frac{2n+1}{2} } }}
}}
==References==
<references/>
{{Requires references|grade=D|msg=Tough one, it's just known!}}
{{Definition|Elementary things|Pre-requisites of mathematics|Analysis|Real Analysis}}