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Magazine Chapter 17: Problem 11
Express the given function in the form \(f(z)=u+i v\) $$ f(z)=z^{3}-4 z $$
Short Answer
Expert verified
\( f(z) = (x^3 - 3xy^2 - 4x) + i(3x^2y - y^3 - 4y) \).
Step by step solution
01
Rewrite the Function
Given the function \( f(z) = z^3 - 4z \), where \( z = x + yi \) with \( x, y \) being real numbers, we need to find both the real part \( u \) and the imaginary part \( v \). First, substitute \( z = x + yi \) into the function: \( f(z) = (x + yi)^3 - 4(x + yi) \).
02
Expand \( (x + yi)^3 \)
Use the binomial theorem to expand \( (x + yi)^3 \): \[ (x + yi)^3 = x^3 + 3x^2(yi) + 3x(yi)^2 + (yi)^3 \] Since \( i^2 = -1 \) and \( i^3 = -i \), this expands further to: \[ (x + yi)^3 = x^3 + 3x^2yi - 3xy^2 - y^3i \] Which simplifies to: \[ = (x^3 - 3xy^2) + i(3x^2y - y^3) \].
03
Simplify \(-4(x + yi)\)
Now expand and simplify \(-4(x + yi) = -4x - 4yi\). This expression in terms of real and imaginary parts becomes: \( = -4x + i(-4y) \).
04
Combine and Simplify Terms
Combine the results from Steps 2 and 3: \[ f(z) = (x^3 - 3xy^2) + i(3x^2y - y^3) - 4x - i4y \] Combine the real parts and imaginary parts separately: Real part: \( u = x^3 - 3xy^2 - 4x \) Imaginary part: \( v = 3x^2y - y^3 - 4y \).
05
Present Final Expression
Thus, the expression of \( f(z) = u + iv \) is \[ f(z) = (x^3 - 3xy^2 - 4x) + i(3x^2y - y^3 - 4y) \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
The binomial theorem is a crucial tool for expanding powers of binomials, like \((x + yi)^3\), into sums involving terms of the form \(a^n b^{k-n}\). In this exercise, we apply it to \((x + yi)^3\), where \(x\) and \(yi\) are treated as the two separate terms of a binomial.
This expansion helps us break down the function into manageable parts using a formula: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \], where \(\binom{n}{k}\) is a binomial coefficient.
For \((x + yi)^3\):
This expansion helps us break down the function into manageable parts using a formula: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \], where \(\binom{n}{k}\) is a binomial coefficient.
For \((x + yi)^3\):
- Use binomial coefficient 3 for each term.
- Calculate each term: \(x^3, 3(x^2)(yi), 3(x)(yi)^2, (yi)^3\).
Real and Imaginary Parts
Every complex number can be expressed as a sum of a real part and an imaginary part. Given a complex number \(z = x + yi\), the real part is \(x\) and the imaginary part is \(yi\). This principle extends to complex functions like \(f(z)\), where \(f(z) = u + iv\).
Here, \(u\) and \(v\) are the real and imaginary parts of the function after substituting \(z = x + yi\) into it.
Here, \(u\) and \(v\) are the real and imaginary parts of the function after substituting \(z = x + yi\) into it.
- The function \(f(z) = (x^3 - 3xy^2 - 4x) + i(3x^2y - y^3 - 4y)\) shows how complex expressions split into these parts.
- Real part \(u\) involves only \(x\) and \(xy^2\) terms, while the imaginary part \(v\) consists of terms with \(x^2y\) and \(y\).
Complex Numbers
Complex numbers are numbers composed of a real part and an imaginary part. They are typically represented in the form \(z = x + yi\), where \(x\) is the real component and \(yi\) is the imaginary component.
The imaginary part involves the unit \(i\), defined by the property \(i^2 = -1\).
The imaginary part involves the unit \(i\), defined by the property \(i^2 = -1\).
- This exercise utilizes complex numbers by substituting \(z\) with \(x + yi\), allowing functions to be manipulated in terms of their real and imaginary parts.
- This manipulation is key in finding solutions to functions defined over complex numbers.
Polynomial Expansion
Polynomial expansion is the method of expressing a power of a polynomial as a sum of terms. In the exercise, \((x + yi)^3\) is expanded to its polynomial form using the binomial theorem.
The process involves multiplying each term according to the power it's raised; here, expanding \((x + yi)^3\) results in:
The process involves multiplying each term according to the power it's raised; here, expanding \((x + yi)^3\) results in:
- \(x^3\) being the straightforward expansion of the base term.
- Cross-multiplying terms to produce combinations, like \(3x^2(yi)\) and removing imaginary unit complexities such as substituting \(i^2\) with \(-1\).
- Combining all results gives a clear distinction between what constitutes the real versus imaginary components.
