# Exponential Form of e^z for z = 4e^(i*pi/3)

• MaxManus
In summary, to write e^z in the form a + bi, we first convert z to the form a + bi by using Euler's formula, z = r(cos theta + i*sin theta). Plugging in the given values, we get z = 4*(cos(pi/3) + i*sin(pi/3)), which simplifies to z = 2 + 2*sqrt(3). Then, we can write e^z as e^(2)*(cos(2*sqrt(3)) + i*sin(2*sqrt(3))), which is the solution provided.
MaxManus

## Homework Statement

write e^z in the form a +bi
z = 4e^(i*pi/3)

---------------------------------------
My guess:

z = 4*(cos(pi/3) + i*sin(pi/3))

e^z = e^(4*(cos(pi/3) + i*sin(pi/3))) = e^(4*cos(pi/3))*(cos(4*sin(pi/3)) + i*sin(4*sin(pi/3)))

but the solution says

e^(2)*(cos(2*sqrt(3)) + i*sin(2*sqrt(3)))

MaxManus said:

## Homework Statement

write e^z in the form a +bi
z = 4e^(i*pi/3)

---------------------------------------
My guess:

z = 4*(cos(pi/3) + i*sin(pi/3))

e^z = e^(4*(cos(pi/3) + i*sin(pi/3))) = e^(4*cos(pi/3))*(cos(4*sin(pi/3)) + i*sin(4*sin(pi/3)))

but the solution says

e^(2)*(cos(2*sqrt(3)) + i*sin(2*sqrt(3)))

Well, $\sin(\pi/3)=\frac{\sqrt{3}}{2}$, and I'm sure you know what $\cos(\pi/3)$ is...

Thanks for the help.
--------------------------------------
z = 4e^(i*pi/3)
z = 4*(cos(pi/3) + i*sin(pi/3))
z = 2 + 2*sqrt(3)
e^z = e^(2)*(cos(2*sqrt(3)) + i*sin(2*sqrt(3))

## 1. What is the exponential form of e^z for z = 4e^(i*pi/3)?

The exponential form of e^z is written as e^z = e^(x+iy) = e^x * e^(iy) where z = x+iy. In this case, z = 4e^(i*pi/3) can be written as e^(4+4i*pi/3) = e^4 * e^(4i*pi/3).

## 2. What is e?

e is a mathematical constant approximately equal to 2.71828. It is also known as Euler's number and is commonly used in mathematical and scientific calculations.

## 3. What is z?

z is a complex number with a real part and an imaginary part, written as z = x+iy. In the given equation, z = 4e^(i*pi/3), where x = 4 and y = pi/3.

## 4. What is the significance of pi/3 in the exponential form of e^z for z = 4e^(i*pi/3)?

The term pi/3 represents the angle in radians in the polar form of a complex number. In this case, it indicates the direction of the complex number z = 4e^(i*pi/3) on the complex plane.

## 5. How is the exponential form of e^z for z = 4e^(i*pi/3) used in practical applications?

The exponential form of e^z is used in various fields of science and engineering, such as electrical engineering, physics, and statistics. It is particularly useful in solving differential equations, representing complex numbers, and describing oscillatory phenomena.

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