Is j^{-p}=e^{-j\frac{p\pi}{2}} a valid exponential function?

yungman · 2013-10-10T09:38:06+00:00

Homework Statement I want to verify j^{-p}=e^{-j\frac{p\pi}{2}} Homework Equations e^{j\frac{\pi}{2}}=\cos(\frac{\pi}{2})+j\sin(\frac{\pi}{2})=j The Attempt at a Solution j^{-p}=(e^{j\frac{\pi}{2}})^{-p}=e^{-j\frac{p\pi}{2}} Am I correct? Thanks

Thread starter yungman
Start date Oct 10, 2013
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Exponential Exponential function Function

In summary, an exponential function is a mathematical function where the independent variable appears in the exponent. It is commonly written in the form y = ab^x and is used to model growth or decay over time. To verify if a function is exponential, one can check if it follows the general form y = ab^x or plot it on a graph. The main difference between an exponential function and a power function is the placement of the independent variable. Exponential functions can be used for both positive and negative values, but for negative values, the function must have a fraction as the exponent. Exponential functions are used in real life for various purposes such as population growth, compound interest, and modeling natural phenomena.

Oct 10, 2013

yungman

Homework Statement

I want to verify
[tex]j^{-p}=e^{-j\frac{p\pi}{2}}[/tex]

Homework Equations

[tex]e^{j\frac{\pi}{2}}=\cos(\frac{\pi}{2})+j\sin(\frac{\pi}{2})=j[/tex]

The Attempt at a Solution

[tex]j^{-p}=(e^{j\frac{\pi}{2}})^{-p}=e^{-j\frac{p\pi}{2}}[/tex]

Am I correct?
Thanks

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Oct 10, 2013

dirk_mec1

Yes.

1. What is an exponential function?

An exponential function is a mathematical function in which the independent variable appears in the exponent. It is commonly written in the form y = ab^x, where a and b are constants. Exponential functions are often used to model growth or decay over time.

2. How do you verify if a function is exponential?

To verify if a function is exponential, you can check if it follows the general form y = ab^x. Additionally, you can plot the function on a graph and see if it forms a curved line that increases or decreases rapidly. Another way is to calculate the ratio of y-values for a constant change in x-values. If the ratio is constant, the function is exponential.

3. What is the difference between an exponential function and a power function?

The main difference between an exponential function and a power function is the placement of the independent variable. In an exponential function, the independent variable appears in the exponent, while in a power function, it appears as the base. Another difference is that exponential functions have a constant ratio of y-values for a constant change in x-values, while power functions have a constant difference between y-values for a constant change in x-values.

4. Can exponential functions be used for both positive and negative values?

Yes, exponential functions can be used for both positive and negative values. However, for negative values, the function must have a fraction as the exponent. For example, y = 2^(-x) is an exponential function with a negative exponent, and it will produce a curve that decreases rapidly as the x-values increase.

5. How are exponential functions used in real life?

Exponential functions are used in various real-life situations, such as population growth, compound interest, and radioactive decay. They can also be used to model the spread of diseases and the growth of bacteria. In finance, exponential functions are used to calculate future values of investments. In physics, they are used to model natural phenomena like radioactive decay and heat transfer.