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Chapter 2: Problem 3

Solve for \(z\) : (a) \(z^{5}=1\) (b) \(3+2 e^{z-i}=1\) (c) \(\tan z=1\)

Short Answer

Expert verified
The solutions are: (a) \( z_k = e^{2\frac{\text{蟺} i k}{5}} \) for \( k = 0, 1, 2, 3, 4 \). (b) \( z = i(\text{蟺} + 1) \). (c) \( z = \frac{\text{蟺}}{4} + n\text{蟺}, \) where \( n \) is any integer.

Step by step solution

01

Solve the equation \(z^5 = 1\)

To solve the equation \(z^5 = 1\), recognize that this is a complex number problem where we need to find the 5th roots of unity. The general formula for the roots is given by: \[ z_k = e^{2\frac{\text{蟺} i k}{5}} \text{ for } k = 0, 1, 2, 3, 4 \]Thus, the five solutions are: \[ z_0 = e^{2 \frac{\text{蟺} i \times 0}{5}} = 1 \]\[ z_1 = e^{2 \frac{\text{蟺} i \times 1 }{5}} \]\[ z_2 = e^{2 \frac{\text{蟺} i \times 2}{5}} \]\[ z_3 = e^{2 \frac{\text{蟺} i \times 3 }{5}} \]\[ z_4 = e^{2 \frac{\text{蟺} i \times 4}{5}} \]
02

Simplify \(3 + 2e^{z-i} = 1\)

Simplify and solve the given equation for \(z\). Start by subtracting 3 from both sides: \[ 3 + 2e^{z-i} = 1 \ 2e^{z-i} = -2 \ e^{z-i} = -1 \]Next, take the natural logarithm of both sides:\[ z - i = \text{ln}(-1) \ z - i = i\text{蟺} (since \text{ln}(-1) = i\text{蟺}) \ z = i\text{蟺} + i \ z = i(\text{蟺} + 1) \]
03

Solve \( \tan z = 1 \)

The equation \( \tan z = 1 \) is solved by identifying the general solutions for the tangent function. Recall that \[ \tan z = 1 \text{ at } z = \frac{\text{蟺}}{4} + n\text{蟺} \text{ for } n \text{ as any integer} \ \text{Thus, the solutions are:} \ z = \frac{\text{蟺}}{4} + n\text{蟺}, \text{ where } n \text{ is any integer.} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Roots of Unity
Understanding the concept of 'roots of unity' is crucial for solving complex number equations like the one given in part (a) of the exercise. Simply put, the roots of unity are the solutions to the equation \[ z^n = 1 \]where \(n\) is a positive integer. These roots are situated uniformly on the unit circle in the complex plane and can be expressed using Euler's formula. For instance, the 5th roots of unity are given by: \[ z_k = e^{2\frac{\pi i k}{5}} \]where \(k\) ranges from 0 to 4. This formula helps us find all the distinct roots which, in this case, are:
  • \( z_0 = e^{2 \frac{\pi i \times 0}{5}} = 1 \)
  • \( z_1 = e^{2 \frac{\pi i \times 1 }{5}} \)
  • \( z_2 = e^{2 \frac{\pi i \times 2}{5}} \)
  • \( z_3 = e^{2 \frac{\pi i \times 3 }{5}} \)
  • \( z_4 = e^{2 \frac{\pi i \times 4}{5}} \)
Knowing these roots is particularly helpful in solving polynomial equations in complex numbers.
Complex Logarithm
The 'complex logarithm' comes into play when dealing with complex exponentiation, as seen in part (b) of the exercise. To solve the equation \(3 + 2e^{z-i} = 1\), we first isolate the exponential term:\[ 2e^{z-i} = -2 \]Now, taking the natural logarithm of both sides helps us proceed:\[ z - i = \text{ln}(-1) \]In the realm of complex numbers, the natural logarithm of \(-1\) is particularly interesting because it equals \(i\pi\):\[ z - i = i\pi \]Solving for \(z\) yields:\[ z = i(\pi + 1) \].This example illustrates how understanding the complex logarithm is essential for solving such equations and how it applies to Euler's formula and the unit circle.
Trigonometric Functions
Trigonometric functions are fundamental in solving various types of equations, particularly those involving the tangent function, like in part (c) of the exercise. Recall that the tangent function has periodic solutions:\[ \tan z = 1 \]This equation is true when:\[ z = \frac{\pi}{4} + n\pi \]where \(n\) is any integer. This periodicity means that the solutions will occur at regular intervals of \(\pi\) along the complex plane. Breaking it down:
  • \( z = \frac{\pi}{4} + 0\pi = \frac{\pi}{4} \)
  • \( z = \frac{\pi}{4} + 1\pi = \frac{5\pi}{4} \)
  • \( z = \frac{\pi}{4} + 2\pi = \frac{9\pi}{4} \)
These solutions demonstrate the importance of understanding how trigonometric functions behave within the complex plane and their applications in complex number equations.

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Most popular questions from this chapter

Determine whether the following functions are analytic. Discuss whether they have any singular points or if they are entire. (a) \(\tan z\) (b) \(e^{\sin z}\) (c) \(e^{1 /(z-1)}\) (d) \(e^{\bar{z}}\) (e) \(\frac{z}{z^{4}+1}\) (f) \(\cos x \cosh y-i \sin x \sinh y\)

Let \(f(z)\) be analytic in some domain. Show that \(f(z)\) is necessarily a constant if either the function \(\overline{f(z)}\) is analytic or \(f(z)\) assumes only pure imaginary values in the domain.

Given the complex analytic function \(\Omega(z)=z^{2}\), show that the real part of \(\Omega, \phi(x, y)=\operatorname{Re} \Omega(z)\), satisfies Laplace's equation, \(\nabla_{x, y}^{2} \phi=0\). Let \(z=(1-w) /(1+w)\), where \(w=u+i v\). Show that \(\phi(u, v)=\operatorname{Re} \Omega(w)\) satisfies Laplace's equation \(\nabla_{u, v}^{2} \phi=0\).

Let \(f(z)\) be an entire function, with \(|f(z)| \leq C|z|\) for all \(z\), where \(C\) is a constant. Show that \(f(z)=A z\), where \(A\) is a constant.

Consider two entire functions with no zeroes and having a ratio equal to unity at infinity. Use Liouville's Theorem to show that they are in fact the same function.
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