Complex Analysis (i ,immediately)

In summary: If so, then you've found the complex root of z^3=1. For 2), the cosine and sine of a complex number are real, so the two sides must be real too. If you minus one of the cosines, you get a real number. So the two sides must be equal.
  • #1
phykb
2
0
Please help me with them problems:
1) if z^3=1, show that (1-z)(1-z^2)(1-z^4)(1-z^5)=9, zEC

2) if cos(x)+cos(y)+cos(t)=0, sin(x)+sin(y)+sin(t)=0 show that cos(3x)+cos(3y)+cos(3t)=3cos(x+y+t)

3)show that, the roots the equations (1+z)^(2n) +(1-z)^(2n)=0, nEN, zEC are given by the relation z=itan((2κ+1)π)/4n), κ=0,1,...,n-1
 
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  • #2
phykb said:
Please help me with them problems:
1) if z^3=1, show that (1-z)(1-z^2)(1-z^4)(1-z^5)=9, zEC

2) if cos(x)+cos(y)+cos(t)=0, sin(x)+sin(y)+sin(t)=0 show that cos(3x)+cos(3y)+cos(3t)=3cos(x+y+t)

3)show that, the roots the equations (1+z)^(2n) +(1-z)^(2n)=0, nEN, zEC are given by the relation z=itan((2κ+1)π)/4n), κ=0,1,...,n-1

You must show some effort of your own, before we can offer tutorial help. How would you start working on each of these?
 
  • #3
berkeman said:
You must show some effort of your own, before we can offer tutorial help. How would you start working on each of these?

Problem (1), (3): i have no idea
Problem (2): question...cos(x)cos(y)cos(t)-cos(x)sin(y)sin(t)-sin(x)cos(y)sin(t)-sin(x)sin(y)cos(t)=cos(y) (cos(t) cos(x)-sin(t) sin(x))-sin(y) (sin(t) cos(x)+cos(t) sin(x))=cos(x+y+t)? if answer=yes, I'm ok with this problem, i solve this problem!

Please help me with the problems (1), (2)
 
  • #4
phykb said:
Problem (1), (3): i have no idea
Problem (2): question...cos(x)cos(y)cos(t)-cos(x)sin(y)sin(t)-sin(x)cos(y)sin(t)-sin(x)sin(y)cos(t)=cos(y) (cos(t) cos(x)-sin(t) sin(x))-sin(y) (sin(t) cos(x)+cos(t) sin(x))=cos(x+y+t)? if answer=yes, I'm ok with this problem, i solve this problem!

Please help me with the problems (1), (2)
You mean problems 1 and 3.

For 1, if z3 = 1, then z must be one of the complex cube roots of 1. All three have magnitude 1, but different args (angles). One of them has an arg of 2pi/3. Maybe you can come up with the other two.

For each one, evaluate (1-z)(1-z2)(1-z4)(1-z5), and see what you get. That's how I would approach it.

For 3, I don't have any insights right now, but I would start playing with it. For example, I would check that z = i tan(1/4) is a solution of your equation. (This is the solution for K = 0.)
 
  • #5
For 1) there are three complex roots of z^3=1, as Mark44 said. But z=1 doesn't work since (1-z)=0. So you must mean one of the other ones. But there is a simple way to do it. Since z^3=1, z^4=z and z^5=z^2. So you've now got (1-z)^2*(1-z^2)^2. Since |z|=1 and z(z^2)=1 that means z^2=z* (*=complex conjugate). Now you've got ((1-z)(1-z*))^2. If you expand that one of the parts is z+z*. Can you show that's (-1) for z a complex root of z^3=1?
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that studies the properties and behavior of complex numbers, which are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. It deals with functions and equations involving complex numbers, and their derivatives and integrals.

2. What is the difference between real analysis and complex analysis?

The main difference between real analysis and complex analysis is that real analysis deals with functions and equations involving real numbers, while complex analysis deals with functions and equations involving complex numbers. Additionally, the techniques and tools used in each branch are different, as complex numbers have unique properties that require a different approach to solve problems.

3. What is the importance of complex analysis?

Complex analysis has many applications in mathematics and other fields such as physics, engineering, and economics. It is used to solve problems involving electrical circuits, fluid dynamics, signal processing, and quantum mechanics. It also provides a deeper understanding of real analysis and helps in solving complex problems in other areas of mathematics.

4. What are some key concepts in complex analysis?

Some key concepts in complex analysis include analytic functions, Cauchy-Riemann equations, contour integration, and the Cauchy integral theorem. Other important topics include power series, Laurent series, and residues, which are used to evaluate complex integrals. Complex analysis also involves the study of complex mappings, such as conformal mappings and Mobius transformations.

5. Are there any real-world applications of complex analysis?

Yes, there are many real-world applications of complex analysis. For example, in electrical engineering, complex analysis is used to analyze AC circuits and solve problems involving impedance and frequency response. In physics, it is used to study electromagnetic fields and quantum mechanics. Complex analysis also has applications in finance, signal processing, and computer graphics.

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