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.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]
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.
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.
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.
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.
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.