Continuing my blogging adventures at the Last Drop Cafe in Claremont, CA:
CauchyReimann Equations
Let the complexvalued function f(z), where z = x+i*y, be defined in the parametric form:
f(z) = u(x,y) + i*v(x,y) where u(x,y) = Re(f(z)) and v(x,y) = Im(f(z)).
Then by the definition of the derivative:
f(z) = lim x0 → x [(f(z) + x0)  f(z))/(x  x0)]
= lim x0 → x [(u(x +x0, y)  u(x, y) + i*v(x + x0, y)  i*v(x, y))/(x  x0)]
= du/dx + i*dv/dx (I)
Also:
f(z) = lim y0 → y [(f(z + y0)  f(z))/(i*y  i*y0)]
= lim y0 → y [(u(x, y + y0)  u(x, y) + i*v(x, y + y0)  i*v(x, y))/(i*(y  y0))]
= i*(lim y0 → y [(u(x, y + y0)  u(x, y) + i*v(x, y + y0)  i*v(x, y))/(y  y0)])
= i*du/dy + dv/dy (II)
Taking the real and imaginary parts of (I) and (II):
du/dx = dv/dy and du/dy = dv/dx
The CauchyRiemann equations can be used to determine whether f(z) is differentiable and if so, where.
Two Quick Examples
1. z^2 = (x^2  y^2) + 2*i*x*y
u=x^2  y^2
v=2*i*x*y
du/dx = 2*x, dv/dy = 2*x
du/dy = 2*y, dv/dx = 2*y (note the negative sign in front of du/dy!)
Since the CauchyRiemann equations hold, and without restriction, then z^2 is differentiable for all z. And:
df/dz = 2*x + 2*i*y = 2*(x+i*y) = 2*z
2. e^z = e^(x+i*y) = e^x*cos y + i*e^x*sin y
u=e^x*cos y
v=e^x*sin y
du/dx = e^x*cos y, dv/dy = e^x*cos y
du/dy = e^x*sin y, dv/dx = e^x*sin y
Since the CauchyRiemann equations hold, and without restriction, then e^z is differentiable for all z. And:
df/dx = e^x*cos y + i*e^x*sin y = e^z
I used (I), but using (II) will garner the same result.
That is the CauchyRiemann equations in a nutshell!
Source: Wunsch, A. David. Complex Variables with Applications. 2nd Edition. AddisonWesley Publishing Company. 1994.
As always, have a great day! Take care. Eddie
This blog is property of Edward Shore. 2014
A blog is that is all about mathematics and calculators, two of my passions in life.
Saturday, November 22, 2014
Complex Analysis: CauchyReimann Equations
Subscribe to:
Post Comments (Atom)
Some Algebra Word Problems
Some Algebra Word Problems Source: Blitzer, Robert Introductory & Intermediate Algebra for College Students 3 rd Edition Pearso...

TI36X Pro Review This is a review of the TI36X Pro Calculator by Texas Instruments. History Originally, this was the TI30X Pro that w...

Casio fx991EX Classwiz Review Casio FX991EX The next incarnation of the fx991 line of Casio calculators is the fx991 EX. ...

One of the missing features of the TI82/83/84 family is the ability to convert between bases. Here are two programs in TIBasic to help...
No comments:
Post a Comment