What is Weierstrass equation?
A Weierstrass equation or Weierstrass model over a field k is a plane curve E of the form y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , y^2 + a_1xy + a_3y = x^3 + a_2 x^2 + a_4 x + a_6, y2+a1xy+a3y=x3+a2x2+a4x+a6, with a 1 , a 2 , a 3 , a 4 , a 6 ∈ k a_1, a_2, a_3, a_4, a_6 \in k a1,a2,a3,a4,a6∈k.
Is the Weierstrass function a fractal?
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
Can you integrate the Weierstrass function?
The antiderivative of the Weierstrass function is fairly smooth, i.e. not too many sharp changes in slope. This just means that the Weierstrass function doesn’t rapidly change values (except in a few places). integrals, unlike derivatives, are highly insensitive to small changes in the function.
Is Weierstrass function uniformly continuous?
The Weierstrass function f from Equation (1) is everywhere continuous. an cos(bnπx) converges uniformly to f(x) on R.
Is the Weierstrass function periodic?
It is periodic with period 2π. You can see it’s pretty bumpy. So bumpy, in fact, that it’s not differentiable anywhere.
What are the equations for the circle center?
If we place the circle center at (0,0) and set the radius to 1 we get: (x−a) 2 + (y−b) 2 = r 2 (x−0) 2 + (y−0) 2 = 1 2 x2 + y2 = 1
Which is the easiest equation to make a circle?
A circle is easy to make: Draw a curve that is “radius” away There are an infinite number of those points, here are some examples: In all cases a point on the circle follows the rule x 2 + y 2 = radius 2 Because it may not be in the neat “Standard Form” above. (x-a) 2 + (y-b) 2 = r 2 (x-0) 2 + (y-0) 2 = 1 2
Can you write a circle equation in general form?
Because it may not be in the neat “Standard Form” above. It is a circle equation, but “in disguise”! So when you see something like that think “hmm that might be a circle!” In fact we can write it in “General Form” by putting constants instead of the numbers: Note: General Form always has x2 + y2 for the first two terms.
Which is the formula for the radius of a circle?
Radius = r = 1 Standard form of circle equation is (x – a) 2 + (y – b) 2 = r 2 Substituting the values of centre and radius, (x – 2) 2 + (y – 3) 2 = 1 2