# Proving Zero Result with Complex Conjugates and Dot Product

• Scootertaj
In summary, the task is to prove that \int_{-\infty}^{+\infty} \! i*(\overline{d/dx(sin(x)du/dx})*u \, \mathrm{d} x + \int_{-\infty}^{+\infty} \! \overline{u}*(d/dx(sin(x)du/dx) \, \mathrm{d} x equals 0. The work done so far involves using the fact that \overline{u} is the complex conjugate of u and * represents the dot product. However, the correctness of the work is uncertain.
Scootertaj

## Homework Statement

Show that the following = 0:
$\int_{-\infty}^{+\infty} \! i*(\overline{d/dx(sin(x)du/dx})*u \, \mathrm{d} x + \int_{-\infty}^{+\infty} \! \overline{u}*(d/dx(sin(x)du/dx) \, \mathrm{d} x$ where $\overline{u}$ = complex conjugate of u and * is the dot product.

2. Work so far
My thoughts: $\int_{-\infty}^{+\infty} \! i*(\overline{d/dx(sin(x)du/dx})*u \, \mathrm{d} x + \int_{-\infty}^{+\infty} \! \overline{u}*(d/dx(sin(x)du/dx) \, \mathrm{d} x$
=
$\int_{-\infty}^{+\infty} \! -i*(d/dx(sin(x)du/dx)*u \, \mathrm{d} x + \int_{-\infty}^{+\infty} \! \overline{u}*(d/dx(sin(x)du/dx) \, \mathrm{d} x$

But I don't even know if that's right.

Bump.

## 1. What is the definition of an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is denoted by the symbol ∫ and is used to calculate the total value of a function over a given interval.

## 2. What is a complex conjugate?

A complex conjugate is a mathematical term that refers to a pair of complex numbers with the same real part but opposite imaginary parts. For example, the complex conjugate of 3+4i is 3-4i.

## 3. How are integrals and complex conjugates related?

The integral of a complex conjugate function is equal to the complex conjugate of the integral of the original function. This means that the order of integration and complex conjugation can be switched without affecting the final result.

## 4. What is the importance of complex conjugates in mathematics?

Complex conjugates are important in many areas of mathematics, including complex analysis, signal processing, and quantum mechanics. They are used to simplify equations and solve problems involving complex numbers.

## 5. Can complex conjugates be used in the integration of complex functions?

Yes, complex conjugates can be used in the integration of complex functions. They can help to simplify integrals and make them easier to solve, especially when dealing with functions involving complex numbers.

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