Bringing a derivative inside an integral?

In summary, in order to show that a function f(z) is analytic by showing f'(z) exists, one must use the Leibniz rule to bring the derivative inside the contour integral. This may not work the same for contour integrals, and there are certain conditions that must be satisfied by g(z,\zeta). These conditions are explained in more detail in the link provided.
  • #1
AxiomOfChoice
533
1
I'm trying to show that a function [itex]f(z)[/itex] is analytic by showing [itex]f'(z)[/itex] exists. But [itex]f(z)[/itex] is defined in terms of a contour integral:
[tex]
f(z) = \oint_{|\zeta - z_0| = r} g(z,\zeta) d\zeta.
[/tex]
Since the integral is being carried out with respect to [itex]\zeta[/itex] and not [itex]z[/itex], am I allowed to bring the [itex]d/dz[/itex] operator inside the integral? Or is it more complicated than that? Are there certain conditions that [itex]g(z,\zeta)[/itex] must satisfy? If so, what are they?

THANKS!
 
Physics news on Phys.org
  • #3
I think you might want to use leibnitz rule.
 

What is the purpose of bringing a derivative inside an integral?

The purpose of bringing a derivative inside an integral is to simplify the expression and make it easier to solve. It allows us to use the rules of differentiation to evaluate the integral.

Can any derivative be brought inside an integral?

No, only certain types of derivatives can be brought inside an integral. These include derivatives of polynomials, trigonometric functions, exponential functions, and logarithmic functions.

How do you bring a derivative inside an integral?

To bring a derivative inside an integral, you can use the substitution method or the integration by parts method. Substitution involves replacing the variable in the integral with a new variable that is equal to the derivative of the original variable. Integration by parts involves using the product rule of differentiation to rewrite the integral.

What are the benefits of bringing a derivative inside an integral?

Bringing a derivative inside an integral can make it easier to evaluate the integral, especially if the integral is complex or involves multiple variables. It also allows us to use our knowledge of derivatives to solve integrals.

Are there any limitations to bringing a derivative inside an integral?

Yes, there are limitations to bringing a derivative inside an integral. It can only be done for certain types of derivatives, and it may not always result in a simpler expression. Additionally, the substitution or integration by parts method may not always work for every integral.

Similar threads

I Cauchy Integral Formula with a singularity
    • Calculus
Replies
21
Views
2K
I On a remark regarding the Cauchy integral formula
    • Topology and Analysis
Replies
2
Views
635
I What is the value of the integral for higher order poles in the real axis?
    • Calculus
Replies
9
Views
2K
I Fractional Integral of which function is equal to Riemann's Zeta-Function?
    • Topology and Analysis
Replies
2
Views
1K
I Derivation of Crooks's Fluctuation Theorem
    • Classical Physics
Replies
3
Views
1K
I What if the Jacobian doesn't exist at finite points in domain of integral?
    • Calculus
Replies
1
Views
1K
A Summing over continuum and uncountable numerocities
    • Calculus
Replies
1
Views
928
A Derivation of a complex integral with real part
    • Calculus
Replies
5
Views
1K
A Help needed with derivation: solving a complex double integral
    • General Math
Replies
1
Views
739
MHB Calculate integral using Stokes Theorem
    • Calculus
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top