Introduction to Complex Functions

Summary

In this video lecture, the instructor introduces the concept of functions of complex variables, starting with the basics of mapping complex numbers to other complex numbers. The video explores real and imaginary parts of complex functions and dives into the criteria for differentiability, specifically focusing on holomorphic functions. Through examples, such as z^2 and cubic functions, the instructor illustrates the strict requirements for a function to be differentiable in the complex plane. The video concludes by examining the differences between differentiable and non-differentiable complex functions, laying the groundwork for further discussions on complex calculus and contour integration.

Takeaways

• 😀 A complex function maps a complex number (Z) to another unique complex number (W), which also has real and imaginary parts.
• 😀 A complex number has both a real and an imaginary component. The function's output is similarly composed of a real part (u) and an imaginary part (v).
• 😀 For a function of a complex variable, the real and imaginary parts of the output are functions of the real and imaginary parts of the input.
• 😀 The example function Z^2 (where Z = x + yi) is expanded to show how the real and imaginary parts can be separated.
• 😀 Holomorphic functions are those that are differentiable at every point in a region, with unique derivatives at each point.
• 😀 The derivative of a complex function is defined similarly to real functions, using the limit of Delta F / Delta Z as Delta Z approaches zero.
• 😀 For complex functions, the derivative must exist in all directions (left, right, up, down, diagonally), making differentiability a stricter condition than for real functions.
• 😀 A real function can only approach a point from the left or right, but a complex function can approach from any direction in the complex plane.
• 😀 Z^3 is shown to be a holomorphic function because its derivative is the same regardless of the direction from which the limit is approached.
• 😀 The function f(Z) = 2y + xi is shown not to be holomorphic because its derivative depends on the direction from which Z approaches zero (e.g., from the x or y axis).
• 😀 For a function to be holomorphic, its derivative must be the same in all directions, which is not the case for the example f(Z) = 2y + xi.

Q & A

What is a function of a complex variable?

A function of a complex variable is a relation that maps a complex number Z to another unique complex number W. The output W is also a complex number, composed of a real part and an imaginary part.

How are the real and imaginary parts of a complex function determined?

The real and imaginary parts of a complex function are derived from the real and imaginary parts of the input Z. For instance, if Z = x + yi, the function W can be expressed as W = u(x, y) + v(x, y)i, where u and v are functions of x and y.

Can you provide an example of how a complex function can be written in terms of real and imaginary parts?

Yes, for example, for the function f(Z) = Z^2, where Z = x + yi, the expanded form is (x + yi)^2 = x^2 - y^2 + 2xyi. Thus, the real part u(x, y) is x^2 - y^2, and the imaginary part v(x, y) is 2xy.

What is the goal when working with complex functions?

The goal is to perform calculus on complex functions, including differentiation and integration (specifically contour integration). For these functions to be differentiable, they must have continuous and well-defined derivatives.

What does it mean for a complex function to be holomorphic?

A complex function is holomorphic in a region R of the complex plane if it has a unique derivative at every point in R. This means that the function is differentiable from all directions in the complex plane, not just from the left or right.

How is the derivative of a complex function defined?

The derivative of a complex function f(Z) is defined similarly to real functions. It is the limit of the change in the function (Δf) over the change in the variable (ΔZ) as ΔZ approaches zero.

What makes the conditions for differentiability stricter for complex functions compared to real functions?

In real functions, the input variable can approach a point from only two directions (left and right). However, for complex functions, the input can approach a point from any direction in the complex plane, including diagonals. For a function to be differentiable, the derivative must be the same from all directions.

What is the significance of having equal derivatives from all directions in a complex function?

For a complex function to be differentiable (holomorphic), the derivatives from all directions must be equal. If the derivatives differ depending on the direction, the function is not differentiable at that point.

Can you give an example of a differentiable complex function?

Yes, for example, the cubic function f(Z) = Z^3 is differentiable everywhere in the complex plane. Its derivative is f'(Z) = 3Z^2, which is continuous and well-defined in all directions.

What is an example of a non-differentiable complex function?

An example is the function f(Z) = 2y + xi, where Z = x + yi. The derivatives along different directions (such as the x-axis and y-axis) yield different values, indicating that the function is not differentiable at any point and is not holomorphic.