Problem 30 - Advanced Engineering Mathematics

Question

Write \(f(z)=\operatorname{Im}(z-3 \bar{z})+z \operatorname{Re}\left(z^{2}\right)-5 z\) in the form \(f(z)=u(x, y)+i v(x, y)\)

Step-by-Step Solution

Answer

\(f(z) = (x^3 - xy^2 - 5x) + i(4y + yx^2 - y^3 - 5y)\)

Step 1: Express the Components in Terms of x and y

Start with the complex number expression. Let \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part of \(z\). The conjugate \(\bar{z} = x - yi\). Substitute these into the original function: \(f(z) = \operatorname{Im}(z - 3\bar{z}) + z \operatorname{Re}(z^2) - 5z\).

Step 2: Simplify the Imaginary Part Calculation

Calculate \(z - 3\bar{z} = (x + yi) - 3(x - yi) = x + yi - 3x + 3yi = -2x + 4yi\). The imaginary part \(\operatorname{Im}(-2x + 4yi) = 4y\).

Step 3: Simplify the Second Term Calculation

Find \(z^2 = (x + yi)^2 = x^2 + 2xyi - y^2\). The real part is \(\operatorname{Re}(z^2) = x^2 - y^2\). Therefore, \(z \operatorname{Re}(z^2) = (x + yi)(x^2 - y^2) = x(x^2 - y^2) + yi(x^2 - y^2) = x^3 - xy^2 + i(yx^2 - y^3)\).

Step 4: Simplify the Third Term

Calculate \(-5z = -5(x + yi) = -5x - 5yi\). This splits into \(-5x\) for the real part and \(-5y\) for the imaginary part.

Step 5: Combine All Components to Find Re(u) and Im(v)

Combine the results: - Real part: \(u(x, y) = 0 + x^3 - xy^2 - 5x\)- Imaginary part: \(v(x, y) = 4y + yx^2 - y^3 - 5y\)Thus, \(f(z) = (x^3 - xy^2 - 5x) + i(4y + yx^2 - y^3 - 5y)\)

Key Concepts

Imaginary Part

When dealing with complex numbers, the imaginary part refers to the coefficient of the imaginary unit, usually denoted as "i," in the expression. In simpler terms, if you have a complex number, written as \(z = x + yi\), the imaginary part is the \(y\) in this equation.

• The imaginary part is crucial in handling complex numbers, as it lets us measure how far a number deviates from being purely real.
• The imaginary unit \(i\) might at first seem abstract, but it follows specific mathematical rules, particularly \(i^2 = -1\).

One interesting note on the imaginary part, from the exercise, is the calculation of \(z - 3\bar{z}\). This calculation gave us \(-2x + 4yi\), and the imaginary part was shown to be simply \(4y\). This operation illustrates how abstract complex operations can still reveal tangible, meaningful results.

Complex Conjugate

The complex conjugate of a complex number \( z = x + yi \) is an essential concept in complex analysis. It is derived by changing the sign of the imaginary component, which leads to \( \bar{z} = x - yi \).

• Utilizing conjugates helps simplify expressions involving complex numbers.
• Conjugates also play a fundamental role in dividing complex numbers.

In the exercise, we use the complex conjugate to simplify the expression \(z - 3\bar{z}\). By finding the conjugate of \(z\), we can express this operation and showcase how the complex conjugate assists in simplifying complex numbers under various algebraic manipulations, as seen in reducing the equation to \(-2x + 4yi\).

Complex Numbers

Complex numbers combine real and imaginary numbers into a single entity to solve equations that ordinary numbers cannot. When expressing a complex number, it usually takes the form \(z = x + yi\), where \(x\) (Real Part) and \(yi\) (Imaginary Part) create a robust numerical system able to tackle a wide array of problems.

• Complex numbers unlock solutions to equations like \(x^2 + 1 = 0\), thanks to the imaginary unit \(i\).
• They broaden the number scope from the real number line to the complex plane.
• Every complex number can be fully represented through persistence with both its real and imaginary parts.

The exercise at hand exemplifies complex analysis through deriving functions as sums of products and transformations of real and imaginary coefficients, illustrating the beauty and flexibility of complex numbers.

Real Part

The real part of a complex number \(z = x + yi\) is the component \(x\) that isn't associated with the imaginary unit \(i\). This part can be considered analogous to regular numbers on a number line, separate from any imaginary components.

• The real part keeps the power of additional dimensions in complex analysis anchored to known, understandable values.
• In polar and exponential representations of complex numbers, the real part is critical for defining magnitude and practical applications.

In the exercise's calculation of \(z \operatorname{Re}(z^2)\), the real part was found to be \(x^2 - y^2\). Upon substitution in the function, this part profoundly influences the overall expression, serving as a reminder of how the real part merges with other terms, granting complex numbers their extensive applicability.