# How exp(jwt)=cos(wt)+jsin(wt)?

• nomisme
So as t increases, the exponential will increase indefinitely along the real axis, but will repeat along the imaginary axis, never reaching infinity. This is why exp(jwt) is sinusoidal. In summary, the conversation discusses the behavior of exp(jwt) and its relation to sine and cosine functions. It is explained that exp(jwt) is sinusoidal due to its periodic behavior along the imaginary axis and that it can be represented using Taylor series and compared to cosine and sine functions. The importance and usefulness of understanding this concept is also highlighted.
nomisme
Can anyone explain in simple terms on how they are equal? graphs are welcome.

The thing I don't understand is why exp(jwt) is sinusoidal. Wouldn't exp(jwt) go to infinity when t increases? I don't know how to understand the exp term as sinusoidal? in my mind they simply go to zero or infinity.

Assuming j is the imaginary unit just compare the Taylor series for these two functions.

nomisme said:
Can anyone explain in simple terms on how they are equal? graphs are welcome.

The thing I don't understand is why exp(jwt) is sinusoidal. Wouldn't exp(jwt) go to infinity when t increases? I don't know how to understand the exp term as sinusoidal? in my mind they simply go to zero or infinity.

exp(x) behaves the way you expect. The image of exp(ix) is the unit circle on the complex plane.

notice they solve the same differential equation.

1 person
I like Mathwonk's answer! A similar way to see that $e^{ix}= cos(x)+ isin(x)$ is to use the Taylor's series at x= 0 (as jgens said):

$$e^{x}= 1+ x+ x^2/2+ x^3/6+ \cdot\cdot\cdot+ x^n/n!+ \cdot\cdot\cdot$$
$$cos(x)= 1- x^2/2+ x^4/4!+ \cdot\cdot\cdot+ (-1)^n x^{2n}/(2n!)+ \cdot\cdot\cdot$$
$$sin(x)= x- x^3/3!+ x^5/5!+ \cdot\cdot\cdot+ (-1)^n x^{2n+1}/(2n+1)!+ \cdot\cdot\cdot$$

Each involves powers of x over the factorial of that power. Sine has only odd powers, cosine only even powers and sine and cosine have alternating sign.

Now see what happens when we replace "x" with "ix" in the series for $e^x$:
$$e^{ix}= 1+ (ix)+ (ix)^2/2+ (ix)^3/3!+ \cdot\cdot\cdot+ (ix)^n/n!+ \cdot\cdot\cdot$$
$$e^{ix}= 1+ ix+ i^2x^2/2+ i^3x^3/3!+ \cdot\cdot\cdot+ i^nx^n/n!+ \cdot\cdot\cdot$$
Of course, $i^2= -1$ so $i^3= i^2(x)= -i$, $i^4= -i(i)= 1$ and then it all repeats. That is, every odd power of ix is plus or minus ix while every even power is plus or minus 1. So
$$e^{ix}= 1+ ix- x^2/2- ix^3/3!+ \cdot\cdot\cdot+ (-1)^nx^{2n}/(2n)!+ (-1)^ni x^{2n+1}/(2n+1)!+ \cdot\cdot\cdot$$

and separating "real" and "imaginary" parts
$$e^{ix}= (1- x^2/2+ x^4/4!+ \cdot\cdot\cdot+ (-1)^n x^{2n}/(2n)!+ \cdot\cdot\cdot)+ i(x- x^3/3!+ \cdot\cdot\cdot+ (-1)^n x^{2n+1}/(2n+1)!)$$
$$e^{ix}= cos(x)+ isin(x)$$

I don't know how to understand the exp term as sinusoidal? in my mind they simply go to zero or infinity.
That's because you are used to dealing with real numbers. The exponential is increasing along along the real axis, periodic along the imaginary axis.

## What is the meaning of exp(jwt)?

The term exp(jwt) is used in mathematics and engineering to represent a complex exponential function. It is also known as Euler's formula and is commonly used in fields such as signal processing, quantum mechanics, and electrical engineering.

## How is exp(jwt) related to the trigonometric functions cosine and sine?

Exp(jwt) can be written as cos(wt) + jsin(wt), where w is the angular frequency. This representation shows the relationship between exp(jwt) and the trigonometric functions cosine and sine. The real part, cos(wt), represents the horizontal component, while the imaginary part, jsin(wt), represents the vertical component in a complex plane.

## What is the importance of exp(jwt) in scientific calculations?

Exp(jwt) is important in scientific calculations because it allows us to represent complex numbers in a concise and efficient manner. It also has many applications in fields such as physics, engineering, and mathematics. For example, it is used to describe the behavior of oscillating systems and electromagnetic waves.

## How can exp(jwt) be used in real-life scenarios?

Exp(jwt) can be used in many real-life scenarios, such as modeling the behavior of electrical circuits, analyzing sound waves in music or speech, and understanding the behavior of quantum systems. It is also used in digital signal processing to analyze and manipulate signals in various applications.

## Can exp(jwt) be simplified or expressed in a different form?

Yes, exp(jwt) can be simplified using Euler's formula to e^(jwt), where e is the base of the natural logarithm. It can also be expressed in terms of hyperbolic trigonometric functions, such as cosh(wt) + jsinh(wt). These different forms may be more suitable for specific applications or calculations.

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