The Cauchy-Riemann equations

Publicerat den september 11, 2012 av mattelararen

In order for a complex function of a  single complex variable to be differentiable it must be differentiable both parallell to the imaginary axis δy →0 and parallell to the real axis δx →0.

This condition leads to the CAuchy –Riemann equations-

The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are the two equations:

(1a)     \dfrac{ \partial u }{ \partial x } = \dfrac{ \partial v }{ \partial y } \,

and

(1b)    \dfrac{ \partial u }{ \partial y } = -\dfrac{ \partial v }{ \partial x } \,
Proof:
Suppose that

f(z) = u(z) + i \cdot v(z)

is a function of a complex number z. Then the complex derivative of ƒ at a point z0 is defined by

\lim_{\underset{h\in\mathbb{C}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = f'(z_0)

provided this limit exists.

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds

\lim_{\underset{h\in\mathbb{R}}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \frac{\partial f}{\partial x}(z_0).

On the other hand, approaching along the imaginary axis,

\lim_{\underset{h\in \mathbb{R}}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} =\frac{1}{i}\frac{\partial f}{\partial y}(z_0).

The equality of the derivative of ƒ taken along the two axes is

i\frac{\partial f}{\partial x}(z_0)=\frac{\partial f}{\partial y}(z_0),
Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of \mathbb{C} (this is called a domain in \mathbb{C}). 
 function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.
Publicerat i Advanced, Calculus, Imaginary numbers | Lämna en kommentar

de Moivre’s formula and complex-conjugation.

Publicerat den september 10, 2012 av mattelararen

(e^ix )^n = cos(nx) + i sin(nx) is called de Moivre’s formula.

The formula is named after the 17 th. century French huguenot mathematician Abraham de Moivre.

Also the variable in of a function can be a complex number. f(z) =z^2 and z = x + iy gives x^2 + 2ixy + y^2 with real part x^2 + y*2 and imaginary part 2xy.

A necessary condition for a function of a complex variable to be differentaible is that it satisfies the Cauchy -Riemann equations.

Publicerat i Gymnasiematematik(high school math), Imaginary numbers, matematik 4, matematik 5 | Märkt , , , | Lämna en kommentar

Alternative representations of complex numbers

Publicerat den september 9, 2012 av mattelararen

As mentioned in the latest post any complex number may be represented by an arrow in the complex plane. This number is unambiguously described by two numbers: its real part x and its imaginary part y. z= x+iy. This is called the Cartesian representation. (Rene Descartes)

From the figure below it is evident that

x = r cosφ

y = r sin φ  and thus z = r( cosφ + i sinφ) . This is called polar representation of the complex number z. r is the modulus of the vector z (i.e. its length) and φ is called the argument of z.    The modulus can be  computed by multiplying the  complex number z with its conjugated complex number z

By adding the  Taylor series for cos(x) and  i sin(x) we get the series expansion for the exponential function eix. This relation is called Euler’s formula.

This leads to the exponential representation of a complex number:

z= r e(iφ) 

This

 File:Complex conjugate picture.svg
Publicerat i Imaginary numbers, matematik 4, matematik 5 | Märkt , , , | Lämna en kommentar

Imaginary numbers

Publicerat den september 6, 2012 av mattelararen

A solution to the simple second-degree equation

x2 + 1 =0

can not be found along the line of real-numbers.

Therefore it was necessary to invent a fictive number i such that i2=-1.

i.e. the imaginary numbers making it possible to take the square root of negative numbers.

They are represented along an axis horizontal to the axis of real numbers.

A combination of a real number and a imaginary number is called a complex number.

z = 2  + 3i.

Publicerat i matematik 1c, matematik 4, matematik 5 | Märkt , , , | 2 kommentarer

The first man on the Moon

Publicerat den september 5, 2012 av mattelararen

Full moon

Last week the first human ever to set a foot on the Moon, Neil Armstrong (born 1930) expired.

His family wrote wrote this as a final tribute:

”For those who may ask what they can do to honor Neil, we have a simple request. Honor his example of service, accomplishment and modesty, and the next time you walk outside on a clear night and see the moon smiling down at you, think of Neil Armstrong and give him a wink.”[

Publicerat i Astronomy | Lämna en kommentar

Bessel functions

Publicerat den augusti 30, 2012 av mattelararen

Friedrich Wilhelm Bessel (1784-1846)  was an outstanding mathematician and astronomer in the 19 th. century. Professor at the Albertina university in the no longer existing town of Königsberg. He was the first astronomer to use the parallax of a star for distance measurements. He also pinned down the positions of 50 000 stars.

In pure mathematics his major achievement is to have deduced the Besselfunctions which are solutions to the Bessel differential equation.

x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0
The solutions are given by
J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin \tau) \,\mathrm{d}\tau.
This equation is encountered in electromagnetic wave-propagation problems and in quantum mechanics when solving the Schrödinger-equation.
Publicerat i Calculus | Märkt , | 2 kommentarer

Line integrals

Publicerat den augusti 28, 2012 av mattelararen
The line integral along the curve C can be written as
\int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.
where C is parametrisized as r(t) with the parmeter t.

∫¦r'(t)¦ dt equals the arc length i.e. the length of the curve.

Consider eg the circle.  x2+ y2 = r2. This can be parametrisized as follows

x(t) = cos(t)

y(t) = sin(t)

The perimeter of the circle can then be calculated according to

2π

∫ sint2+cos2t dt= 2π
0

Publicerat i Gymnasiefysik(high school physics), matematik 5 | Lämna en kommentar

Exercises on the masterclass

Publicerat den augusti 22, 2012 av mattelararen

Exercises

Publicerat i Uncategorized | 2 kommentarer

Zeno’s paradoxes

Publicerat den augusti 16, 2012 av mattelararen
Zeno of Elea

It is impossible to arrive at any destination since at first you have to travel half the distance and then half the remaining distance and so on.

This means you have to travel an infinite number of half-distances which ought to take infinitely long time.

Read more about his paradoxes here

Publicerat i Gymnasiematematik(high school math), Uncategorized | Märkt , | 2 kommentarer

Shooting stars

Publicerat den augusti 15, 2012 av mattelararen

Perseid meteor shower

The perseids is a meteor shower connected to the comet Swift-Turtle. This picture shows this year’s Perseid shower photographed over Wirkenheim in Germany.

http://en.wikipedia.org/wiki/Perseids

Publicerat i Astronomy | Lämna en kommentar