Problem 12 - Advanced Engineering Mathematics

Question

Answer Problems 1-16 without referring back to the text. Fill in the blank or answer true/false. If \(f(z)=x^{2}-3 x y-5 y^{3}+i\left(4 x^{2} y-4 x+7 y\right)\), then \(f(-1+2 i)=\) _________.

Step-by-Step Solution

Answer

\(f(-1+2i) = -33 + 26i\).

Step 1: Understand the Function

The function given is complex: \(f(z)=x^{2}-3 x y-5 y^{3}+i\left(4 x^{2} y-4 x+7 y\right)\). Here, \(x\) is the real part, and \(y\) is the imaginary part of the complex number \(z=x+yi\). We need to substitute \(z=-1+2i\) into this function.

Step 2: Substitute Values into the Function

Substitute \(x=-1\) and \(y=2\) into the real and imaginary components of the function: Real part: \((-1)^{2} - 3(-1)(2) - 5(2)^{3}\) Imaginary part: \(4(-1)^{2}(2) - 4(-1) + 7(2)\).

Step 3: Simplify the Real Part

Calculate the real part: \((-1)^{2} = 1\), \(-3(-1)(2) = 6\), \(-5(2)^{3} = -40\). Combine these: \(1 + 6 - 40 = -33\).

Step 4: Simplify the Imaginary Part

Conduct calculations for the imaginary part: \(4(-1)^{2}(2) = 8\), \(-4(-1) = 4\), \(7(2) = 14\). Combine these: \(8 + 4 + 14 = 26\).

Step 5: Combine the Final Result

The function evaluation gives us the result: Real part: \(-33\), Imaginary part: \(26\). Thus, \(f(-1+2i) = -33 + 26i\).

Key Concepts

Complex Functions

Complex functions are mathematical functions that have complex numbers as variables. Unlike real functions, which can be visualized on a two-dimensional graph, complex functions exist in a four-dimensional space because complex numbers have both a real part and an imaginary part. This makes them quite fascinating yet sometimes challenging to grasp.

• In the given exercise, the function is expressed as: \[f(z) = x^2 - 3xy - 5y^3 + i(4x^2y - 4x + 7y)\]
  This represents a complex function where both the real and imaginary components are composed of variables \(x\) and \(y\).
• Understanding how to dissect these components is crucial. The real part is \(x^2 - 3xy - 5y^3\) while the imaginary part is \(4x^2y - 4x + 7y\).
• Evaluating complex functions often requires substituting a specific complex number for \(z\), which consists of a real component \(x\) and an imaginary component \(y\).

Complex functions provide the foundation for more advanced topics such as analytic functions, which delve deeper into properties like continuity and differentiability.

Complex Numbers

Complex numbers are numbers that include both a real and an imaginary part. A complex number can be written in the form of \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part.

• The imaginary part involves the imaginary unit \(i\), which satisfies \(i^2 = -1\).
• In the exercise, the complex number \(-1 + 2i\) is used. Here, the real part is \(-1\) and the imaginary part is \(2\).
• Complex numbers can be represented on the complex plane, with the horizontal axis representing the real part and the vertical axis the imaginary part.

Understanding how to manipulate complex numbers is fundamental in complex analysis, and it involves operations like addition, subtraction, multiplication, and division, all respecting the unique properties of \(i\).

Complex Arithmetic

Complex arithmetic involves performing mathematical operations on complex numbers such as addition, subtraction, multiplication, and division.

• For addition and subtraction, you simply add or subtract the respective real and imaginary components.
• Multiplication of complex numbers employs the distributive property, but it's essential to remember that \(i^2 = -1\). For example, multiplying \(a+bi\) and \(c+di\) gives \((ac-bd) + (ad + bc)i.\)
• For division, we use the concept of the complex conjugate to eliminate the imaginary component from the denominator. This requires multiplying the numerator and the denominator by the conjugate of the denominator.

In the provided exercise, complex arithmetic is utilized to evaluate the function at a particular complex number by performing operations separately on the real and imaginary parts and then combining the results to get the final solution: \(-33 + 26i\). Thus, mastering complex arithmetic is key to solving problems involving complex functions.