Inverse of the sinc function - need to use Ei function?

In summary, the conversation discusses the cancellation of I1 and I2 when t>0.5 or t<-0.5. This is shown by applying a change of variables and does not require the use of the Ei function. There is also confusion about the location of the pole on the real or imaginary axis.
  • #1
thomas49th
655
0

Homework Statement


http://gyazo.com/966f3a03d71843a46a832e6508d6ca95

So when t> 0.5 or t<-0.5 the text says I1 and I2 cancel. Can I show this. When I tried, integration by parts gets me nowhere, so I looked up the integral of form e^x/x and apparently it has "no elementary derivative" - have to use Ei, which I've never heard of. Should I just take it for granted that I1 and I2 cancel?

Also it says about pole being on the real axis, but gives me I1 =i*pi, which is on the imaginary axis. Not quite sure what that part is trying to say

Thanks
Thomas
 
Physics news on Phys.org
  • #2
Try applying the change of variables [itex]\omega'=p_1 \omega[/itex] to I1. What do you notice? And no, you don't need the Ei function.
 

1. What is the inverse of the sinc function?

The inverse of the sinc function is the function that undoes the effects of the sinc function. In other words, it takes the output of the sinc function and returns the input that would produce that output.

2. What is the Ei function?

The Ei function, also known as the exponential integral function, is a special function in mathematics that is defined as the integral of the exponential function from a given value to infinity. It is often used in the study of special functions and has various applications in physics, engineering, and other fields.

3. How is the Ei function related to the inverse of the sinc function?

The inverse of the sinc function can be expressed in terms of the Ei function. Specifically, the inverse of the sinc function can be written as the integral of the Ei function from a given value to infinity. This relationship is important in solving problems involving the inverse of the sinc function.

4. Why is the Ei function necessary to find the inverse of the sinc function?

The inverse of the sinc function cannot be expressed in terms of elementary functions like polynomials, trigonometric functions, etc. Therefore, the Ei function is necessary to find the inverse of the sinc function as it allows us to express it in terms of a special function that can be evaluated numerically.

5. What are some applications of the inverse of the sinc function using the Ei function?

The inverse of the sinc function using the Ei function has various applications in engineering, physics, and signal processing. It is used in the design of filters, in the analysis of signals and systems, and in solving differential equations. It also has applications in optics, acoustics, and other fields where the sinc function is commonly encountered.

Similar threads

Area under an inverse trigonometric function
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Finding the Inverse Function of tanh(x) in the Interval (-1,1)
    • Calculus and Beyond Homework Help
Replies
2
Views
861
Finding marginal distribution of 2d of probability density function
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Arc length of vector function - the integral seems impossible
    • Calculus and Beyond Homework Help
Replies
2
Views
841
Reducing Bessel Function Integral
    • Calculus and Beyond Homework Help
Replies
1
Views
773
A few questions to make sure I understand Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
281
State the domain and range for a given function
    • Precalculus Mathematics Homework Help
Replies
7
Views
391
MATLAB Replacing the "quad" function with "integral" doesn't work in Matlab
    • MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
2K
Can you derive a trigonometric function from its inverse dx?
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
To find the range of a given ##\sin## function
    • Precalculus Mathematics Homework Help
Replies
15
Views
632

Hot Threads

Recent Insights

Back
Top