Evaluate the integral.

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In summary, the conversation discusses a problem involving an integral with complex variables and attempts to find a solution using a formula from a table. It is determined that the formula is valid as long as the real part of one variable is positive and a correction is made for an error in the formula.
  • #1
IHateMayonnaise
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Homework Statement



This is part of a much larger problem, however currently I am stuck at the following integral:

[tex]

\int_{-\infty}^{\infty}exp\left(k^2\left((\Delta x)^2-\frac{i \hbar t}{2m}\right ) + k(ix-2(\Delta x)^2 \bar{k}_x)\right)dk[/tex]

Where obviously everything should be taken as a constant except the plane old k's.

Homework Equations



see (1) and (3)

The Attempt at a Solution



i tried to complete the square, followed by u/du substitution which yielded one of the messiest equations I've ever seen. There is an integral I found in a table that looks promising:

[tex]\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4a}+c\right)[/tex]

However this does not include the imaginary unit therefore I do not believe it is much use to me.

Cookies for anyone who can get me started in the right direction.

Thanks yall!

IHateMayonnaise
 
Last edited:
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  • #2
Your formula with a, b, & c is valid for complex a, b, & c as long as the real part of a is positive.

I think the [itex]k^2(\Delta x)^2[/itex] term in the exponent has the wrong sign.
 
  • #3
Avodyne said:
Your formula with a, b, & c is valid for complex a, b, & c as long as the real part of a is positive.

I think the [itex]k^2(\Delta x)^2[/itex] term in the exponent has the wrong sign.

Why is the formula valid only if the real part of a is positive? (Also, this is assuming that it is a definite integral from [itex]-\infty[/itex] to [itex]\infty[/itex]?)

Good call on the [itex]k^2(\Delta x)^2[/itex] having the wrong sign! Thanks so much :)

Edit: Also I made a mistake in that integral from the table; I originally put:

[tex]\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4ac}+c\right)[/tex]

but I meant:

[tex]\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4a}+c\right)[/tex]
 
Last edited:

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is a fundamental tool in calculus and is used to solve a variety of problems in physics, engineering, and other fields.

What does it mean to "evaluate" an integral?

Evaluating an integral means finding the numerical value of the integral using integration techniques such as the fundamental theorem of calculus, substitution, or integration by parts.

How do I evaluate an integral?

The process of evaluating an integral involves identifying the integrand (the function being integrated), choosing an appropriate integration technique, and applying that technique to find the numerical value of the integral. It may also involve setting limits of integration, which determine the boundaries of the area being calculated.

Why is evaluating integrals important?

Evaluating integrals is important because it allows us to solve a wide range of problems in mathematics, science, and engineering. It is used to calculate areas, volumes, and other quantities that are necessary for understanding and predicting real-world phenomena.

What are the applications of evaluating integrals?

Evaluating integrals has many applications in fields such as physics, engineering, economics, and statistics. It can be used to solve problems involving motion, optimization, probability, and more. It is also an essential tool for understanding and developing theories in these and other areas of study.

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