JB Radius of convergence In order to find the radius of convergence of ...
Radius of convergence
JB In order to find the radius of convergence of any power series, we use either the ratio test or the ratio test, and from these tests we will obtain a value for our Limit L. Radius of convergence is therefore R:= 1/ Limit L Note: The value of the radius of convergence is ALWAYS positive as it is the radius of a disc.
Uniform convergence
We know that f(x) = lim n∞ fn(x) In order to determine whether a function converges uniformly or not, we use the following rules: 1) Supx∈… | fn(x) – f(x) | < Ɛ OR 2) Supx∈… | fn(x) – f(x) | 0 as n ∞
Cauchy-Riemann equations p.145
For a function f(z) of a complex variable z= x + iy, it can also be written in the form: f(x + iy) = u(x,y) + iv(x,y) Using the above format, we need to see whether it satisfies the Cauchy-Riemann equations below: 1) du/dx = dv/dy AND 2) du/dy = - dv/dx or another way of writing the above is: 1) ux = vy 2) uy = - vx
Polar form
AND
Let the complex number be: z= a+bi, where a is the ‘real part’ and b is the ‘imaginary part.’ In order to find the polar form of any complex number, we need to find the modulus and the argument of the complex number (z). 1) Modulus of a complex number (z) is: r= |z| = |a+bi| = √(a2+b2) 2) Argument of a complex number (z) is: arg(z) = arg (a+bi) = an angle known as = tan-1 (b/a) Using the information above, we can obtain the polar form using this rule: z = r(cos +isin) or rei, where ei= cos + isin Corresponding diagram to this complex number:
JB In order to find the radius of convergence of any power series, we use either the ratio test or the ratio test, and from these tests we will obtain a value for our Limit L. Radius of convergence is therefore R:= 1/ Limit L Note: The value of the radius of convergence is ALWAYS positive as it is the radius of a disc.
Uniform convergence
We know that f(x) = lim n∞ fn(x) In order to determine whether a function converges uniformly or not, we use the following rules: 1) Supx∈… | fn(x) – f(x) | < Ɛ OR 2) Supx∈… | fn(x) – f(x) | 0 as n ∞
Cauchy-Riemann equations p.145
For a function f(z) of a complex variable z= x + iy, it can also be written in the form: f(x + iy) = u(x,y) + iv(x,y) Using the above format, we need to see whether it satisfies the Cauchy-Riemann equations below: 1) du/dx = dv/dy AND 2) du/dy = - dv/dx or another way of writing the above is: 1) ux = vy 2) uy = - vx
Polar form
AND
Let the complex number be: z= a+bi, where a is the ‘real part’ and b is the ‘imaginary part.’ In order to find the polar form of any complex number, we need to find the modulus and the argument of the complex number (z). 1) Modulus of a complex number (z) is: r= |z| = |a+bi| = √(a2+b2) 2) Argument of a complex number (z) is: arg(z) = arg (a+bi) = an angle known as = tan-1 (b/a) Using the information above, we can obtain the polar form using this rule: z = r(cos +isin) or rei, where ei= cos + isin Corresponding diagram to this complex number:
