Complex Variables. Complex square root function

In summary: Combining the expressions for f(z) for both vertical and horizontal lines, we get the final form of f(z) = e^(ln√(x^2+y^2)+i*arctan(y/x)/2). This is the equation of a hyperbola, as desired. In summary, we have proven that the function f(z) = √z = e^(lnz/2) maps horizontal and vertical lines in the complex plane, given by the equations z(y) = (a,y) and w(x) = (x,a), respectively, onto hyperbola branches. This is achieved by expressing these lines in terms of the real and imaginary parts of z and
  • #1
ELESSAR TELKONT
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0

Homework Statement



Proof that [tex]f(z)=\sqrt{z}=e^{\frac{\ln z}{2}}[/tex] with logarithm branch [tex][0,2\pi)[/tex]. Then [tex]f[/tex] maps horizontal and vertical lines in [tex]A=\mathbb{C}-\{\mathbb{R}^{+}\cup\{0\}\}[/tex] on hyperbola branches.

Homework Equations



I have that [tex]\ln_{[0,2\pi)} (z)=\ln\vert z\vert+i\mathop{\rm arg}\nolimits_{[0,2\pi)} (z)[/tex]

The Attempt at a Solution



I have tried to proof it directly, that means to describe vertical lines with [tex]z(y)=(a,y)[/tex] and horizontal ones with [tex]w(x)=(x,a)[/tex] and substitute in [tex]f[/tex]. That produces an horrific expression that I can't reduce to an hyperbola. What can I do?
 
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  • #2


Hi there!

Thank you for sharing your attempt at solving this problem. Let me guide you through the proof step by step.

First, let's start by defining the function f(z) = √z = e^(lnz/2). This means that f(z) is the square root of z, where the logarithm branch is [0,2π).

Now, let's consider a vertical line in the complex plane, given by the equation z(y) = (a,y). To map this line onto a hyperbola, we need to express it in terms of the real and imaginary parts of z. Using the definition of z(y), we can write z = x+iy, where x=a and y=y.

Substituting this into our function f(z), we get f(z) = √(x+iy) = e^(ln(x+iy)/2).

Next, we need to express the logarithm ln(x+iy) in terms of the real and imaginary parts of z. Using the identity ln(a+ib) = ln√(a^2+b^2)+i*arctan(b/a), we can write ln(x+iy) = ln√(x^2+y^2)+i*arctan(y/x).

Substituting this into our function f(z), we get f(z) = e^(ln√(x^2+y^2)+i*arctan(y/x)/2).

Now, let's consider a horizontal line in the complex plane, given by the equation w(x) = (x,a). Following the same steps as before, we can write w = x+iy, where x=x and y=a.

Substituting this into our function f(z), we get f(z) = √(x+ia) = e^(ln(x+ia)/2).

Using the same identity as before, we can write ln(x+ia) = ln√(x^2+a^2)+i*arctan(a/x).

Substituting this into our function f(z), we get f(z) = e^(ln√(x^2+a^2)+i*arctan(a/x)/2).

Now, we have expressions for f(z) for both vertical and horizontal lines. To map these lines onto hyperbola branches, we need to express them in
 

1. What are complex variables?

Complex variables refer to numbers that have both real and imaginary components. They can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

2. What is the purpose of studying complex variables?

Complex variables are used to solve problems in various fields of science and engineering, such as physics, chemistry, and electrical engineering. They also have applications in computer science, economics, and finance.

3. What is the complex square root function?

The complex square root function is a mathematical operation that takes the square root of a complex number. It is defined as the inverse of the complex square function, which squares a complex number.

4. How does the complex square root function work?

The complex square root function works by finding the two complex numbers that, when squared, will result in the given complex number. It follows the same rules as the real square root function, where the positive square root is the principal root and the negative square root is the other root.

5. What are some properties of the complex square root function?

Some properties of the complex square root function include:
- It is not defined for negative real numbers
- It has two distinct values for any given complex number
- It preserves the magnitude of a complex number
- It follows the same rules as the real square root function for simplifying expressions

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