Small change in constants, big change in integral

• Rose Garden
In summary, the function 1/(x^2 + a) has three different forms when integrated based on the value of the constant "a": arctan of something for positive a, -1/x for a = 0, and a natural logarithm for negative a. However, these forms are related in a way that a small change in a can lead to a completely different integral that still looks very similar. This is because of the similarities between trigonometric functions and exponentials, and their inverse functions. For example, the derivative of 1/x is related to the natural logarithm, showing the connection between these functions.

Rose Garden

Check this out,
1/(x2+a)
where "a" is a constant

When this function is integrated, if a is positive then we get something like arctan of something, if a is 0 we simply get -1/x, and if a is negative then we get something involving the natural logarithm, and yet there's something very similar to all 3 graphs.

But how is it that a small change in this constant a can lead to such drastic changes in the functional form of the integral?

They are very similar. Generally, you factor the denominator, then use partial fractions. This is what you do when a is negative, or zero. When a is positive, you can do the same thing, but you get complex roots. Then you procede in the normal way and use the relation arctan(z)=(i/2)ln[(1-iz)/(1+iz)] and you should get the same answer.

thanks, I have no problem integrating these, I'm just really amazed by how a change in some constant can lead to integrals of completely different forms, and yet still look very similar, is there any way to explain this?

The similarities are more obvious if you think about functions of complex variables.

$$e^{iz} = \cos z + i \sin z$$

In a sense, "trig functions" and "exponentials" are only different names for the same basic mathematical function.

The same applies to their inverse functions, like log and arctan.

The derivative of 1/x is log x, so in that sense 1/x is also related to the others.

1. What is meant by "small change in constants, big change in integral"?

The phrase refers to a concept in physics where a small change in fundamental physical constants, such as the speed of light or the gravitational constant, can lead to significant changes in the overall behavior of a system.

2. How does this concept relate to the laws of physics?

This concept is related to the laws of physics because it highlights the sensitivity of physical systems to changes in fundamental constants. It shows that even small changes in these constants can have a profound impact on the behavior of the universe.

3. Can you provide an example of "small change in constants, big change in integral"?

One example is the fine-structure constant, which is a dimensionless number that characterizes the strength of the electromagnetic force. A small change in this constant, even by just one part in a billion, would drastically alter the behavior of atoms and molecules, and ultimately have a significant impact on the dynamics of the entire universe.

4. How does this concept affect our understanding of the universe?

This concept challenges our understanding of the universe by showing that even the most fundamental constants are not fixed or absolute. It suggests that there may be a deeper underlying framework that governs these constants and their relationships to each other, which could potentially change our understanding of the laws of physics.

5. Are there any implications of this concept in practical applications?

While the concept of "small change in constants, big change in integral" has mainly been explored in theoretical physics, it has potential implications in practical applications such as cosmology, where it could help explain the observed acceleration of the expansion of the universe. It could also have implications in the development of new technologies, such as quantum computing, where precise control over fundamental constants is crucial.

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