# Complex Analysis Properties Question

• RJLiberator
In summary, by using properties, it can be shown that (sqrt(5)-i) can be expressed as sqrt((sqrt(5))^2+(-1)^2), which equals sqrt(6). Similarly, (2zbar+5) can be represented as (2z+5). However, there seems to be a typo in the problem as sqrt(6)*(2z+5) does not equal sqrt(3)*(2z+5). This suggests that there may be missing absolute value signs. By applying the property that |z+w|^2=|z|^2+|w|^2, it can be shown that sqrt(6)=|sqrt(5)-i|.
RJLiberator
Gold Member
Use properties to show that:
(question is in the attached picture)

Now, it is my understanding that due to properties you can express (sqrt(5)-i) as the sqrt((sqrt(5))^2+(-1)^2) which equals sqrt(6).
And (2zbar+5) can be represented as (2z+5).

But this would be sqrt(6)*(2z+5) which is NOT sqrt(3)*(2z+5)

Was there a typo in this problem? Or am I not thinking of something?

Thank you.

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RJLiberator said:
Use properties to show that:
(question is in the attached picture)

Now, it is my understanding that due to properties you can express (sqrt(5)-i) as the sqrt((sqrt(5))^2+(-1)^2) which equals sqrt(6).
And (2zbar+5) can be represented as (2z+5).

But this would be sqrt(6)*(2z+5) which is NOT sqrt(3)*(2z+5)

Was there a typo in this problem? Or am I not thinking of something?

Thank you.

There are some absolute value signs missing, but yes, I think there is a typo.

RJLiberator
Excellent. After performing the operations, I felt like I had a good understanding of it. This seems to confirm my suspicions :). Kind regards for your help tonight.

RJLiberator said:
Now, it is my understanding that due to properties you can express (sqrt(5)-i) as the sqrt((sqrt(5))^2+(-1)^2) which equals sqrt(6).
What you wrote here is very far from true. The real number ##\sqrt{6}## obviously can't be equal to ##\sqrt{5}-i##, which isn't even real. But Dick said something about missing absolute value signs, so I suppose you could be talking about something like this: For all ##z,w\in\mathbb C##, if ##\operatorname{Re}\bar z w=0##, then ##|z+w|^2=|z|^2+|w|^2##. This implies that ##|\sqrt{5}-i|^2=|\sqrt{5}|^2+|-i|^2=5+1=6##, and this implies that ##\sqrt{6}=|\sqrt{5}-i|##.

RJLiberator

## What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It involves the analysis of complex-valued functions, which are functions that take complex numbers as inputs and outputs.

## What are the properties of complex analysis?

Some of the key properties of complex analysis include the Cauchy-Riemann equations, the Cauchy integral theorem, the Cauchy integral formula, and the residue theorem. These properties are used to study the behavior of complex functions and their derivatives.

## How is complex analysis used in real life?

Complex analysis has many applications in various fields such as physics, engineering, and economics. In physics, it is used to study electromagnetic fields and fluid dynamics. In engineering, it is used in the design of electrical circuits and control systems. In economics, it is used in the study of market dynamics and financial mathematics.

## What are the main techniques used in complex analysis?

The main techniques used in complex analysis include contour integration, power series, Laurent series, and conformal mapping. These techniques enable mathematicians to solve complex problems involving complex functions and their properties.

## What are some common examples of complex analysis problems?

Some common examples of complex analysis problems include finding the roots of complex polynomials, evaluating integrals of complex functions, and determining the behavior of complex functions at different points. Other applications include solving differential equations involving complex functions and studying the geometry of complex numbers.

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