Help Evaluating Integral with Euler's Formula

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In summary, the conversation discusses the topic of evaluating a Fourier transform without tables, specifically focusing on an integral involving exponential and trigonometric functions. The individual has attempted two methods - integration by parts and using Euler's formula - but has not been successful. They inquire about the possibility of finding the integral in a table and receive a suggestion to integrate by parts twice to get an equation involving the original integral.
  • #1
jaygatsby
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This is not a homework problem, but a problem in the textbook that is not required. I am doing this to get a handle on the topic.

I am evaluating a Fourier transform, without tables, and need to evaluate this integral:

[tex]
\int e^{-t} * sin(2 \pi f_c t) * e^{-j2 \pi ft} dt
[/tex]

I have tried two methods: 1) integration by parts, and 2) integration after expressing the sine function as a complex exponentials. I get stuck in both cases.

The asterisks are there to assist with clarity of spacing. Thanks for any help you can provide,
J
 
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  • #2
Use Euler formula to get exp(-t)*trig function. This is a standard integral (find in table).

Trig function: sin(at), integral = a/(1 + a2)
cos(at), integral = 1/(1 + a2)
(a > 0 for both)
 
  • #3
Thanks, I did try Euler's formula but then worked the integral out manually (attempted to...)

So this integral I would find in the table exclusively, and never try without a table? The way the drill is stated in the book (not a homework problem.), I wonder if I am to work it out without a table.

Thanks,
J
 
  • #4
You can integrate by parts twice to get an equation involving the original integral.

I(exp(-t)cos(at)) = 1 + aI(exp(-t)sin(at)) = 1 - a2I(exp(-t)cos(at))

Similarly for sin(at) integral.
 
  • #5
Thank you
 

FAQ: Help Evaluating Integral with Euler's Formula

What is Euler's formula and how is it used to evaluate integrals?

Euler's formula, also known as the Euler identity, is a mathematical equation that relates complex numbers to trigonometric functions. It is expressed as e^(ix) = cos(x) + i*sin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is the angle in radians. This formula can be used to simplify complex integrals involving trigonometric functions, making them easier to evaluate.

Can Euler's formula be used to evaluate any type of integral?

No, Euler's formula is specifically useful for evaluating integrals involving trigonometric functions. It may not be applicable to other types of integrals, such as polynomial or exponential functions.

Do I need to know complex numbers to use Euler's formula for evaluating integrals?

Yes, since Euler's formula involves the use of imaginary numbers, it is important to have a basic understanding of complex numbers in order to apply it in integral evaluation. However, for simpler integrals, it is possible to use the real part of the formula and avoid complex numbers altogether.

Are there any limitations to using Euler's formula for integral evaluation?

While Euler's formula can be a useful tool in simplifying complex integrals, it may not always provide the most efficient method of evaluation. In some cases, using other integration techniques such as substitution or integration by parts may be more efficient.

Can Euler's formula be used to evaluate definite integrals?

Yes, Euler's formula can be used to evaluate both indefinite and definite integrals. When evaluating definite integrals, the limits of integration should be substituted into the formula, and the resulting complex number can then be simplified to find the final answer.

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