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

Short Answer

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

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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 $$ \frac{\cos(2\ln(\pi))}{\pi} + i\frac{\sin(2\ln(\pi))}{\pi} $$

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.

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