Complex fuction - is it analytic

[leila]

Ok, here is the question

determine whether f(z)=(1+z)/(1-z) is analytic or otherwise. Unfortunetly I am having problems with the maths. So far I have substituted z=x+iy and got

1+x-iy/1-x-iy and if i let a=x+1 and b=1-x then that simplifies my problem to

a+iy/b-iy

so now i have to rearrange this fraction so that i have a term with i and a term without.

For the life of me I can't figure out how to actually do that

Any help would be much appreciated

Leila

 

[HallsofIvy]

Ok, here is the question
determine whether f(z)=(1+z)/(1-z) is analytic or otherwise. Unfortunetly I am having problems with the maths. So far I have substituted z=x+iy and got
1+x-iy/1-x-iy and if i let a=x+1 and b=1-x then that simplifies my problem to
a+iy/b-iy
so now i have to rearrange this fraction so that i have a term with i and a term without.
For the life of me I can't figure out how to actually do that
Any help would be much appreciated
Leila

Analytic where? f(z) is not even defined at z= 1 and so cannot be analytic there. It is a simple rational function and so is analytic everywhere except at z= 1.

Apparently you are trying to use the Cauchy-Riemann equations to show that. To convert a fraction such as (a+iy)/(b-iy) multiply both numerator and denominator by the complex conjugate of the denominator:
$$\frac{a+iy}{b-iy}= \frac{a+iy}{a-iy}\frac{a+iy}{a+iy}= \frac{a^2+ 2ayi- y^2}{a^2+ y^2}= \frac{a^2- y^2}{a^2+y^2}+ \frac{2ay}{a^2+ y^2}i$$.

 

[Architeuthis Dux]

Hi Leila,

Thank you for reaching out about this problem. To determine if a complex function is analytic, we need to check if it satisfies the Cauchy-Riemann equations. These equations state that the partial derivatives of the real and imaginary parts of the function must exist and be continuous at a given point.

In this case, we have f(z) = (1+z)/(1-z), where z = x+iy. If we let a = x+1 and b = 1-x, then we can rewrite the function as f(z) = (a+iy)/(b-iy). Now, to check if this function is analytic, we need to calculate the partial derivatives of the real and imaginary parts of f(z) with respect to x and y.

Real part: Re(f(z)) = (a^2+b^2)/(a^2+b^2) = 1

Imaginary part: Im(f(z)) = (2ay)/(a^2+b^2)

Taking the partial derivatives, we get:

∂Re(f(z))/∂x = 0

∂Im(f(z))/∂y = (2a^2-2b^2)/(a^2+b^2)^2

∂Re(f(z))/∂y = 0

∂Im(f(z))/∂x = (2a^2-2b^2)/(a^2+b^2)^2

Since these partial derivatives do not match, the Cauchy-Riemann equations are not satisfied and therefore, f(z) is not analytic.

I hope this helps! Let me know if you have any other questions.

Best,