# How Does Euler's Identity Simplify the Expression y = e^(x(1-i)) + e^(x(1+i))?

• jaejoon89
In summary, The conversation discusses the equation y = e^(x(1-i)) + e^(x(1+i)) and how it can be simplified to y = (e^x)sinx + (e^x)cosx using Euler's identity. It is also mentioned that the expansion 2cos(x)e^x is not equal to sin(x)e^x + cos(x)e^x.
jaejoon89
How does y = e^(x(1-i)) + e^(x(1+i))
work out to y = (e^x)sinx + (e^x)cosx?

Using Euler's identity I get,

y = (e^x)e^-ix + (e^x)e^ix
y = e^x(cosx - isinx + cosx + isinx)
y = e^x(2cosx)

why don't you write your equations using [tex] tabs...
it will be more readable by others...

jaejoon89 said:
How does y = e^(x(1-i)) + e^(x(1+i))
work out to y = (e^x)sinx + (e^x)cosx?
It doesn't.

jaejoon89 said:
How does y = e^(x(1-i)) + e^(x(1+i))
work out to y = (e^x)sinx + (e^x)cosx?
Obviously, by Euler's identity, (e^x)sinx+ (e^x)cosx= e^(x(1+i)) only, not that sum.

HallsofIvy said:
Obviously, by Euler's identity, (e^x)sinx+ (e^x)cosx= e^(x(1+i)) only, not that sum.

Halls meant (e^x)*i*sinx+ (e^x)cosx. The expansion you did, 2cos(x)e^x, is correct. and it's not equal to sin(x)e^x+cos(x)e^x.

## 1. What is Euler's identity?

Euler's identity, also known as Euler's formula, is a mathematical equation that relates five fundamental mathematical constants: e (the base of the natural logarithm), π (pi), i (the imaginary unit), 1 (the multiplicative identity), and 0 (the additive identity). It is written as e^(iπ) + 1 = 0.

## 2. Who discovered Euler's identity?

Euler's identity was discovered by the Swiss mathematician Leonhard Euler in the 18th century. He is considered one of the most influential mathematicians in history, and his contributions to various fields of mathematics, including calculus and number theory, have had a lasting impact on modern mathematics.

## 3. What is the significance of Euler's identity?

Euler's identity is significant because it connects three of the most important mathematical operations - addition, multiplication, and exponentiation - with three fundamental mathematical constants. It has been called "the most beautiful equation in mathematics" and is often used as an example of the elegance and simplicity of mathematical concepts.

## 4. How is Euler's identity used in real life?

Euler's identity has many practical applications in fields such as physics, engineering, and signal processing. It is also used in complex analysis and number theory, and has implications in quantum mechanics and electrical engineering.

## 5. Is Euler's identity always true?

Yes, Euler's identity is always true. It has been proven mathematically and has been rigorously tested and verified through countless calculations and applications. It is a fundamental truth in mathematics, and any attempt to disprove it would require a fundamental change in our understanding of numbers and operations.

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