Help with these two problems in complex analysis

In that case, one must be careful to specify which branch of the inverse of the exponential is being used, etc.)
  • #1
Mathematicsss

Homework Statement


What is the argument of -4-3i, and -4+3i?

Homework Equations


tantheta=opposite/adjacent side
The principle of argument is that the argument lies between -pi and pi (not including -pi).

The Attempt at a Solution


arg(-4-3i) = -pi + arctan(3/4)
arg(-4+3i) = pi - arctan(3/4)

My teacher wrote on the answer sheet that the argument of -4-3i is just arctan(3/4).. am I incorrect in the above arguments?[/B]
 
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  • #2
You are correct.
 
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Likes scottdave
  • #3
Drawing a picture also will help you figure out an approximate angle, to see if you figured it correctly.
 
  • #4
Mathematicsss said:

Homework Statement


What is the argument of -4-3i, and -4+3i?

Homework Equations


tantheta=opposite/adjacent side
The principle of argument is that the argument lies between -pi and pi (not including -pi).

The Attempt at a Solution


arg(-4-3i) = -pi + arctan(3/4)
arg(-4+3i) = pi - arctan(3/4)

My teacher wrote on the answer sheet that the argument of -4-3i is just arctan(3/4).. am I incorrect in the above arguments?[/B]
Your prof. may have been referring to the fact that angles in the Complex plane depend on the "frame of reference" for angles, as well as to the periodicity. If , e.g., the x-axis corresponds to 0 , then you will have a certain angle, if you set the y-axis to be the 0 -reference, you will have another angle, etc. This relates to what is called a branch of the associated function of logarithm.
 

FAQ: Help with these two problems in complex analysis

What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers, which are numbers that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). It involves the analysis of functions of complex variables, which are functions that take complex numbers as inputs and outputs. Complex analysis has applications in various fields such as physics, engineering, and economics.

What are some common concepts in complex analysis?

Some common concepts in complex analysis include analytic functions, Cauchy-Riemann equations, contour integration, and the Cauchy integral theorem. Analytic functions are functions that are differentiable at every point in their domain and can be represented by power series. The Cauchy-Riemann equations describe the conditions for a function to be analytic. Contour integration is a method used to evaluate integrals along a path in the complex plane. The Cauchy integral theorem states that the value of a contour integral is only dependent on the values of the function inside the contour.

What are some applications of complex analysis?

Complex analysis has numerous applications in various fields such as physics, engineering, economics, and even art. In physics, it is used to study electromagnetism and fluid dynamics. In engineering, it is used in the design of electrical circuits and signal processing. In economics, it is used in the study of stock market trends. In art, it is used to create intricate and beautiful fractal patterns.

What are the two problems in complex analysis?

The two main problems in complex analysis are the Riemann hypothesis and the Poincaré conjecture. The Riemann hypothesis is a famous unsolved problem in mathematics that states that all non-trivial zeros of the Riemann zeta function lie on a specific line in the complex plane. The Poincaré conjecture, which was solved in 2003, states that any simply connected, closed three-dimensional manifold is topologically equivalent to a three-dimensional sphere.

What are some helpful resources for learning complex analysis?

Some helpful resources for learning complex analysis include textbooks such as "Complex Analysis" by Lars Ahlfors and "Visual Complex Analysis" by Tristan Needham. Online resources such as video lectures on platforms like YouTube and Coursera can also be useful. Additionally, seeking out a mentor or joining a study group can greatly aid in understanding complex analysis concepts.

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