Re{} and Im{} operators under the integral sign

In summary, the conversation is about swapping the Re (or Im) operator and the integral sign, and whether the dummy variable should be real or complex. It is clarified that if the complex function has a real variable, then the two expressions are equivalent, but if the function is complex in both the variable and the integral, this may not be the case.
  • #1
eliotsbowe
35
0
Hello, I'm trying to figure out what hypothesis I need to swap the Re{} (or Im) operator and the integral sign, but I can't find anything on the matter. I guess either it's a trivial question or a rare one. Can someone help me?

Thanks in advance.
 
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  • #2
is the dummy variable real or complex?
 
  • #3
If your complex function is f(t)+ g(t)i where t is a real variable, and the integral is, say, [itex]\int Re(f)dt= \int f(t)dt[/itex], then, yes, that is the same as [itex]Re \int f(t)+ g(t)i dt[/itex] because that last integer is [itex]\int f(t)dt+ i\int g(t)dt[/itex].

If, however, f is a complex function of a complex variable that is not necessarily true.
 
  • #4
thanks a lot!
 

Related to Re{} and Im{} operators under the integral sign

1. What is the purpose of using the Re{} and Im{} operators under the integral sign?

The Re{} and Im{} operators are used in complex analysis to separate the real and imaginary parts of a complex number or function. When used under the integral sign, they allow for the integration of complex-valued functions.

2. How do the Re{} and Im{} operators affect the integration process?

The Re{} and Im{} operators do not change the integral itself, but rather they change how the integrand is evaluated. By separating the real and imaginary parts, the integral can be broken down into simpler components for evaluation.

3. Are there any limitations to using the Re{} and Im{} operators under the integral sign?

The Re{} and Im{} operators can only be used for integrands that are complex-valued functions. They cannot be used for real-valued functions or for integrals with real limits of integration.

4. Can the Re{} and Im{} operators be used in all types of integrals?

Yes, the Re{} and Im{} operators can be used in all types of integrals, including indefinite, definite, and improper integrals. As long as the integrand is a complex-valued function, the operators can be applied to simplify the integration process.

5. Is there any other notation that can be used instead of the Re{} and Im{} operators under the integral sign?

Yes, sometimes the notation "Re[]" and "Im[]" is used instead of the Re{} and Im{} operators. Both notations serve the same purpose of separating the real and imaginary parts of a complex-valued function for integration.

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