Hand written solution of complex Analysis chp#1

 Hand written solution of chp#1

Complex analysis is a branch of mathematics that focuses on the study of complex numbers and their properties. Complex numbers consist of a real part (a) and an imaginary part (bi), where 'a' and 'b' are real numbers and 'i' represents the imaginary unit, √(-1). This branch of mathematics explores the behavior and properties of functions that operate on complex numbers.

topics includes:

complex analysis and their properties,complex number, polar form of complex number, power and Roots, set's of point in the complex plane.

Complex numbers can be represented in polar form, which expresses them in terms of their magnitude (r) and argument (θ). The magnitude of a complex number represents its distance from the origin on the complex plane, while the argument represents the angle between the positive real axis and the line connecting the origin to the point.

Complex numbers can be raised to powers and have roots, similar to real numbers. When raising a complex number to a power, the magnitude is raised to that power, and the argument is multiplied by the power. Finding the nth root of a complex number involves taking the nth root of its magnitude and dividing its argument by n.

hand written solution of complex analysis chp#1

In complex analysis, sets of points in the complex plane are often studied. These sets can represent regions that satisfy specific equations or inequalities. The shapes of these regions can vary, including disks, annuli, or more complex patterns Studying complex analysis provides a deeper understanding of functions and their behavior on the complex plane. It allows for the analysis of properties like differentiability, integration, and mappings between different regions. Complex analysis finds applications in fields such as physics, engineering, computer science, and finance.

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In the upcoming chapters, we will delve into complex functions, contour integration, Taylor series, and other fascinating topics within complex analysis. Through this exploration, we will gain insights into the intricate world of mathematics and its practical applications. Stay tuned for Chapter 2, where we will explore complex functions and their properties.