Integration with fraction in exponential

In summary, the conversation is about an integral problem with a fraction in the exponential equation. The function is not integrable in elementary functions and various methods have been tried without success. It is mentioned that this is not a homework problem and is a part of someone's work.
  • #1
Elsasw
10
0
Hi all, am stuck with the integral with fraction in exponential

The equation

I=∫exp(bz)/(a+iz)*exp[(6*a^2-ik*w^2*z)/(z^2+a^2)]*dz

I already tried to partial fraction the 2nd exponential term, then i tried to perform integration by parts but it doesn't work well.i tried substitution too

Thank you.
 
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  • #2
The function [itex]f(z) = \exp\left(bz +\frac{6a^2-ik\omega^2z}{z^2+a^2} \right)/(a+iz)[/itex] is not integrable in the elementary functions. In other words, you can try every standard integration trick in the books and you will not come up with a closed form solution for this integral.

About the best you can do is to expand this function as a power series and integrate term by term.Where did you come up with this problem? If this is homework you may have made a mistake somewhere along the way. Math homework problems typically don't involve functions that aren't integrable in the elementary functions, at least not until you start working with special functions.
 
  • #3
Thanks DH..Nop..it is not homework...its a part of my work...but thanks to you or i will be still trying to integrate it in terms of elementary function.
 

1. What is the general rule for integrating fractions with exponential functions?

The general rule for integrating fractions with exponential functions is to first rewrite the fraction as a power of the exponential function. Then, integrate the exponential function using the power rule and divide by the power. For example, the integral of 1/x^2 can be rewritten as x^-2, and using the power rule we get (-1/(2x)) + C.

2. How do I handle fractions with negative exponents when integrating?

To handle fractions with negative exponents, we can use the power rule in reverse. For example, the integral of x^-2 can be rewritten as 1/(x^2), and using the power rule we get (x^-1)/(-1) + C, which simplifies to -1/x + C.

3. What is the process for integrating a fraction with a variable in the denominator?

The process for integrating a fraction with a variable in the denominator is similar to integrating a regular fraction with an exponential function. First, rewrite the fraction as a power of the variable. Then, use the power rule to integrate the variable and divide by the power. For example, the integral of 1/(x+1) can be rewritten as (x+1)^-1, and using the power rule we get ln(x+1) + C.

4. Can I use substitution to integrate fractions with exponential functions?

Yes, substitution can be used to integrate fractions with exponential functions. This involves substituting a new variable for the exponential function, and then using the appropriate substitution rule for integration. For example, the integral of e^x/(e^x+1) can be rewritten as u/(u+1) using the substitution u=e^x, and then using the substitution rule for integration we get ln(u+1) + C, which simplifies to ln(e^x+1) + C.

5. Are there any special cases to consider when integrating fractions with exponential functions?

Yes, there are a few special cases to consider when integrating fractions with exponential functions. One is when the exponent has a constant added or subtracted, in which case you can use the chain rule to integrate. Another is when the exponent has a coefficient, in which case you can use the power rule and multiply the result by the reciprocal of the coefficient. Additionally, if the fraction has a polynomial in the numerator, you may need to use long division before integrating.

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