# Cauchy's differintegral formula

• Jhenrique
In summary, Cauchy's differintegral formula is a mathematical theorem discovered by the French mathematician Augustin-Louis Cauchy in the early 19th century. It states the relationship between the values of a function and its derivatives and integrals, and is a generalization of the fundamental theorem of calculus. Its significance lies in its practical applications in physics and engineering, particularly in solving differential equations, signal processing, and image reconstruction. Unlike the fundamental theorem of calculus, Cauchy's differintegral formula considers all orders of derivatives and integrals. It is not limited to real numbers, but can also be extended to complex numbers, making it a powerful tool in complex analysis.
Jhenrique
The Cauchy's differintegral formula is: $$\frac{d^n}{dz^n}f(z_0)=\frac{n!}{2\pi i!}\oint_{\gamma}\frac{f(z)}{(z-z_0)^{n+1}}dz$$ But this formula is valid if the derivative is wrt ##\bar{z}## ? $$\frac{d^n}{d\bar{z}^n}f(z_0)$$ And if the integral is wrt ##\bar{z}## is valid too? $$\frac{n!}{2\pi i!}\oint_{\gamma}\frac{f(z)}{(z-z_0)^{n+1}}d\bar{z}$$

Try to prove it!

micromass said:
Try to prove it!

I'm not capable to prove it!

## 1. What is Cauchy's differintegral formula?

Cauchy's differintegral formula is a mathematical theorem that states the relationship between the values of a function and its derivatives and integrals. It is a generalization of the fundamental theorem of calculus.

## 2. Who discovered Cauchy's differintegral formula?

It was discovered by the French mathematician Augustin-Louis Cauchy in the early 19th century.

## 3. What is the significance of Cauchy's differintegral formula?

Cauchy's differintegral formula has many practical applications in physics and engineering, such as solving differential equations, signal processing, and image reconstruction.

## 4. How is Cauchy's differintegral formula different from the fundamental theorem of calculus?

The fundamental theorem of calculus only deals with derivatives and integrals of a function, while Cauchy's differintegral formula considers all orders of derivatives and integrals.

## 5. Is Cauchy's differintegral formula limited to real numbers?

No, Cauchy's differintegral formula can be extended to complex numbers as well, making it a powerful tool in complex analysis.

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