# Real integrals with complex coefficients

• VantagePoint72
In summary, the conversation discusses the validity of using techniques from calculus in one real variable when dealing with complex coefficients. Specifically, the conversation explores whether u-substitution can be used in an integral with complex coefficients and real variables. The result obtained through this technique is confirmed to be correct, but the question remains about the general validity of using familiar tricks with complex numbers.
VantagePoint72
I'm curious about the validity of various techniques from good old calculus in one real variable when dealing with complex coefficients. I know enough complex analysis to know that the rules change when dealing with complex variables, but I'm curious about the case when the variables are still real and its just coefficients that are complex.

For example, suppose you wanted to do the following integral:

$$\int_{-\infty}^{\infty} e^{-\left(t + i\alpha\right)^2} dt$$

with $\alpha$ real. Can I just do u-substitution like this:

$$u = t + i\alpha \rightarrow du = dt \\ \int_{-\infty}^{\infty} e^{-\left(t + i\alpha\right)^2} dt = \int_{u=-\infty}^{u=\infty} e^{-u^2} du$$

and conclude the integral is $\sqrt{\pi}$ like usual? Punching it into Wolfram Alpha confirms that's the result, so it's more whether or not what I did was valid in general that I'm interested in, rather than this particular result. Can I still use the familiar tricks, or does the mere presence of complex numbers invalidate these old techniques?

I am not sure about the validity of the technique but in this case it looks like the right answer. Expand the exponent and get eα2 multiplying the Fourier transform of e-t2, which is e2 (with some constants including √π).

## 1. What are real integrals with complex coefficients?

Real integrals with complex coefficients are mathematical expressions that involve both real numbers and complex numbers. They are usually in the form of definite integrals, where the limits of integration and the integrand (the function being integrated) contain both real and complex numbers.

## 2. How do you solve real integrals with complex coefficients?

The process of solving real integrals with complex coefficients is the same as solving regular integrals, with the addition of some rules for complex numbers. First, the integral is split into two parts - one with real limits and the other with complex limits. Then, the integral with real limits is solved using regular integration techniques, while the integral with complex limits is solved using complex integration techniques such as the Cauchy-Riemann integral formula or contour integration.

## 3. What are the applications of real integrals with complex coefficients?

Real integrals with complex coefficients have various applications in physics, engineering, and other scientific fields. They are used to calculate quantities such as electric fields, magnetic fields, and heat transfer in complex systems. They also play a significant role in solving differential equations and in the study of complex functions.

## 4. Can real integrals with complex coefficients have imaginary values?

Yes, real integrals with complex coefficients can have imaginary values. This is because the limits of integration and the integrand can contain complex numbers, which can result in an imaginary part in the final answer. However, the final result of the integral is always a complex number, which can be written in the form of a real part plus an imaginary part.

## 5. Are there any special techniques for evaluating real integrals with complex coefficients?

There are several special techniques for evaluating real integrals with complex coefficients, such as the Cauchy residue theorem, the Cauchy integral formula, and contour integration. These techniques involve using complex analysis and complex functions to solve the integral. In some cases, these techniques can simplify the integration process and lead to more elegant solutions.

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