Integrating with Substitution: Clarifying the Contour and Chain Rule

[binbagsss]

Homework Statement

Homework Equations

below

The Attempt at a Solution

I have shown that the first identity holds true. Because this is true without it being surrounded by an integral I guess you would need to integrate it all around the same contour $C$. So say I have:

$_C \int \frac{f'(\gamma(t)) d(\gamma(t))}{f(\gamma((t))} = _C \int \frac{f'(t) d(t)}{f(t)} + _C \int \frac{k c}{ct+d} dt$

(Although I am confused because usually when you have what is the left-hand side you have the substitutin rule if $f(x) \to f(y(x))$ then $_{c(x)} \int f(x) dx \to _{c(y(x))} \int f(y(x)) y'(x) dx$ where $c(x)$ is the contour of integration in the original coordinates. (2).)

In which case then does the second expression via applying the substituion integral rule on the left hand side term (as in the first expression) to $c \to \gamma t$ and $f(\gamma t) \to f(t)$ etc. However substitution rule as above (2) I can't seem to understand, I have if $c \to \gamma t$ , $f'(\gamma t) \to f'(\gamma (\gamma(t))$ etc to get $_{\gamma(c)} \int \frac{f'(\gamma(\gamma(t)) d(\gamma(\gamma(t))}{f(\gamma(\gamma(t)))}$

Have I applied the substitution rule wrong or should my starting expression instead be:

$_{\gamma(C)} \int \frac{f'(\gamma(t)) d(\gamma(t))}{f(\gamma((t))} = _C \int \frac{f'(t) d(t)}{f(t)} + _C \int \frac{k c}{ct+d} dt$

many thanks

 

[mighty2000]

Thank you for your post. It seems like you have made some progress in understanding the problem, but there are a few points that need clarification. First, it is important to note that the integration rule you mentioned in (2) only applies when the substitution is a one-to-one function. In this case, the substitution is not one-to-one, so we cannot use that rule directly.

Secondly, when you perform the substitution $c \to \gamma t$, you are essentially changing the contour of integration from $C$ to $\gamma(C)$. This means that the left-hand side of your equation should be written as follows:

$$ _{\gamma(C)} \int \frac{f'(\gamma(t)) d(\gamma(t))}{f(\gamma(t))} = _C \int \frac{f'(\gamma(t)) \gamma'(t) dt}{f(\gamma(t))} $$

Note that the substitution rule is not needed here, as it is already accounted for by changing the contour of integration.

Finally, to make the substitution in the second term, we need to use the chain rule. So the second term should be written as:

$$ _C \int \frac{k c}{ct+d} dt = _C \int \frac{k \gamma(t)}{\gamma(t) t + d} \gamma'(t) dt $$

I hope this helps clarify things. Keep up the good work!