- #1

- 289

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- Homework Statement
- f(z) =(1+z)/(1-z)

- Relevant Equations
- f(z)*=f(z*)

hi guys

i found this problem in a set of lecture notes I have in complex analysis, is the following function real:

$$

f(z)=\frac{1+z}{1-z}\;\;, z=x+iy

$$

simple enough we get

$$

f=\frac{1+x+iy}{1-x-iy}=

$$

after multiplying by the complex conjugate of the denominator and simplification

$$

f=\frac{1-x^{2}-y^{2}}{(1-x)^{2}+y^{2}}+\frac{i2y}{(1-x)^{2}+y^{2}}

$$

clearly the function has an imaginary part, so i assumed its not real!?, but i found in the notes the its a real function by the following proof

$$

f^{*}(z)=\frac{1+z^{*}}{1-z^{*}}=f(z^{*})

$$

how is that a proof that its real?!

i will appreciate any help

i found this problem in a set of lecture notes I have in complex analysis, is the following function real:

$$

f(z)=\frac{1+z}{1-z}\;\;, z=x+iy

$$

simple enough we get

$$

f=\frac{1+x+iy}{1-x-iy}=

$$

after multiplying by the complex conjugate of the denominator and simplification

$$

f=\frac{1-x^{2}-y^{2}}{(1-x)^{2}+y^{2}}+\frac{i2y}{(1-x)^{2}+y^{2}}

$$

clearly the function has an imaginary part, so i assumed its not real!?, but i found in the notes the its a real function by the following proof

$$

f^{*}(z)=\frac{1+z^{*}}{1-z^{*}}=f(z^{*})

$$

how is that a proof that its real?!

i will appreciate any help