/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 use Euler鈥檚 formula to write t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 6

use Euler鈥檚 formula to write the given expression in the form a + ib. $$ \pi^{-1+2 i} $$

Short Answer

Expert verified
Question: Write the given expression $\pi^{-1+2i}$ in the form a + ib. Answer: $\frac{\cos(2\ln(\pi))}{\pi} + i\frac{\sin(2\ln(\pi))}{\pi}$

Step by step solution

01

Write the expression as an exponential function with a base of e

To begin, we need to rewrite the given expression as an exponential function with a base of e. To do this, we can use the property that $$ e^{\ln(a^x)}=a^x $$ for any a and x. Using this property, rewrite the expression: $$ \pi^{-1+2i} = e^{\ln(\pi^{-1+2i})} $$
02

Apply the properties of logarithms

Next, apply the properties of logarithms to simplify the expression further. Specifically, use the properties: $$ \ln(a^b) = b\ln(a) $$ and $$ \ln(a^c d) = c \ln(a) + d \ln(a) $$ So, in our case, $$ e^{\ln(\pi^{-1+2i})} = e^{(-1+2i)\ln(\pi)} $$
03

Apply Euler's Formula

Now that we have the expression in the form of e^(some complex number), we can use Euler's Formula to expand it. $$ e^{(-1+2i)\ln(\pi)} = e^{-\ln(\pi) + 2i\ln(\pi)} = e^{-\ln(\pi)}\cdot e^{2i\ln(\pi)} $$ Applying Euler's Formula, we get: $$ e^{-\ln(\pi)}\cdot e^{2i\ln(\pi)} = e^{-\ln(\pi)}(\cos(2\ln(\pi)) + i\sin(2\ln(\pi))) $$
04

Simplify the expression

Now, let's simplify the expression further using the property $$ e^{\ln(x)} = x $$ We get, $$ e^{-\ln(\pi)}(\cos(2\ln(\pi)) + i\sin(2\ln(\pi))) = \frac{1}{\pi}(\cos(2\ln(\pi)) + i\sin(2\ln(\pi))) $$
05

Write the expression in the form a + ib

Finally, write the expression in the form a + ib. In this case, a is the real part and b is the imaginary part. So, $$ \pi^{-1+2 i} = \frac{1}{\pi}(\cos(2\ln(\pi)) + i\sin(2\ln(\pi))) = \frac{\cos(2\ln(\pi))}{\pi} + i\frac{\sin(2\ln(\pi))}{\pi} $$ The given expression, in the form a + ib, is $$ \frac{\cos(2\ln(\pi))}{\pi} + i\frac{\sin(2\ln(\pi))}{\pi} $$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Complex Numbers
Complex numbers are an essential concept in mathematics, especially in the realm of calculus and electrical engineering. The fundamental unit for complex numbers is the imaginary unit, represented by the letter 'i', where

\( i^2 = -1 \).

A complex number combines a real number and an imaginary number and is written in the form \( a + ib \), where 'a' is the real part, and \( ib \), with 'b' being the imaginary part.

They expand the number system to the complex plane, a two-dimensional space where real numbers lie along the horizontal axis and imaginary numbers along the vertical axis.

Operations with complex numbers follow similar arithmetic rules as real numbers, but with additional considerations for the imaginary unit. For example, when multiplying two imaginary numbers, the result is a real number due to the definition of 'i'. Complex numbers are pivotal in solving polynomials and are used in many fields such as physics, engineering, and signal processing.

Euler's formula is a bridge between trigonometry and complex exponentials, and it states that for any real number x, \( e^{ix} = \text{cos}(x) + i\text{sin}(x) \). This profound relationship is used for expressing complex numbers in polar form and simplifying the multiplication and division of complex numbers.
Exponential Functions
Exponential functions are widely used across various scientific disciplines due to their unique properties. These functions, typically expressed in the form \( f(x) = a^x \), involve a constant base 'a' raised to a variable exponent 'x'.

The number 'e', approximately equal to 2.71828, is a special base for exponential functions. It's the base of the natural logarithm, and exponential functions with this base, \( e^x \), appear frequently in the fields of calculus, physics, and more, due to their behavior with respect to differentiation and integration.

One of the most powerful tools that involves the exponential function with base 'e' is Euler's formula, which relates exponential functions to trigonometric functions. This is key in solving complex arithmetic and calculus problems, particularly when dealing with complex numbers as exponents.

Understanding how exponential functions grow rapidly can also be important in real-world applications, such as in calculating compound interest in finance or understanding population growth in biology.
Properties of Logarithms
Logarithms are the inverse operation of exponentiation and possess properties that make them valuable for simplifying complex mathematical expressions. The logarithm of a number 'x' to a specified base 'a' answers the question: 'To what exponent must 'a' be raised, to produce the number 'x'?'

Common properties of logarithms include:
  • The logarithm of a product equals the sum of the logarithms: \( \text{log}_a(xy) = \text{log}_a(x) + \text{log}_a(y) \).
  • The logarithm of a quotient equals the difference of the logarithms: \( \text{log}_a(\frac{x}{y}) = \text{log}_a(x) - \text{log}_a(y) \).
  • The logarithm of a power equals the exponent times the logarithm: \( \text{log}_a(x^b) = b\text{log}_a(x) \).

These properties allow us to rewrite expressions involving powers and roots in a more manageable form, particularly when we're dealing with exponentiation with complex numbers.

In the field of solving equations, the logarithm is especially advantageous. It helps to linearize exponential relationships, making it easier to isolate variables and solve equations analytically. This is crucial in exponential decay and growth problems, where the 'time' variable is often the exponent in an exponential function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the initial value problem $$ u^{\prime \prime}+\gamma u^{\prime}+u=0, \quad u(0)=2, \quad u^{\prime}(0)=0 $$ We wish to explore how long a time interval is required for the solution to become "negligible" and how this interval depends on the damping coefficient \(\gamma\). To be more precise, let us seek the time \(\tau\) such that \(|u(t)|<0.01\) for all \(t>\tau .\) Note that critical damping for this problem occurs for \(\gamma=2\) (a) Let \(\gamma=0.25\) and determine \(\tau,\) or at least estimate it fairly accurately from a plot of the solution. (b) Repeat part (a) for several other values of \(\gamma\) in the interval \(0<\gamma<1.5 .\) Note that \(\tau\) steadily decreases as \(\gamma\) increases for \(\gamma\) in this range. (c) Obtain a graph of \(\tau\) versus \(\gamma\) by plotting the pairs of values found in parts (a) and (b). Is the graph a smooth curve? (d) Repeat part (b) for values of \(\gamma\) between 1.5 and \(2 .\) Show that \(\tau\) continues to decrease until \(\gamma\) reaches a certain critical value \(\gamma_{0}\), after which \(\tau\) increases. Find \(\gamma_{0}\) and the corresponding minimum value of \(\tau\) to two decimal places. (e) Another way to proceed is to write the solution of the initial value problem in the form (26). Neglect the cosine factor and consider only the exponential factor and the amplitude \(R\). Then find an expression for \(\tau\) as a function of \(\gamma\). Compare the approximate results obtained in this way with the values determined in parts (a), (b), and (d).

The position of a certain spring-mass system satisfies the initial value problem $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+2 u=0, \quad u(0)=0, \quad u^{\prime}(0)=2 $$ (a) Find the solution of this initial value problem. (b) Plot \(u\) versus \(t\) and \(u^{\prime}\) versus \(t\) on the same axes. (c) Plot \(u\) versus \(u\) in the phase plane (see Problem 28 ). Identify several corresponding points on the curves in parts (b) and (c). What is the direction of motion on the phase plot as \(t\) increases?

In the absence of damping the motion of a spring-mass system satisfies the initial value problem $$ m u^{\prime \prime}+k u=0, \quad u(0)=a, \quad u^{\prime}(0)=b $$ (a) Show that the kinetic energy initially imparted to the mass is \(m b^{2} / 2\) and that the potential energy initially stored in the spring is \(k a^{2} / 2,\) so that initially the total energy in the system is \(\left(k a^{2}+m b^{2}\right) / 2\). (b) Solve the given initial value problem. (c) Using the solution in part (b), determine the total energy in the system at any time \(t .\) Your result should confirm the principle of conservation of energy for this system.

find the general solution of the given differential equation. $$ y^{\prime \prime}+2 y^{\prime}+2 y=0 $$

Use the method of variation of parameters to find a particular solution of the given differential equation. Then check your answer by using the method of undetermined coefficients. $$ y^{\prime \prime}-5 y^{\prime}+6 y=2 e^{t} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.