Cauchy’s Integral Formula

Cauchy’s integral formula is used to find or it provides integral formulas for more complex values function with complex variables or holomorphic functions, where holomorphic functions is actually a function with more complex variables and its complex differentiable. 

In simple words the Cauchy’s integral formula could be defined as, a holomorphic function that is defined on a disk is determined by its values on the boundary of the disk, also it provides all the integral formulas for the derivative of that function.

Under Cauchy’s theorem, Cauchy’s integral formula can also be called Cauchy’s differential formula. And it is always an analytic function and is infinitely differentiable.

Cauchy’s Integral formula, A brief history:

• In mathematics, Cauchy’s integral formula was named after Augustin-Louis Cauchy who is a French mathematician and a physicist, is a central statement in complex analysis which in turn is the theory of functions of complex variables.

• Here with Cauchy’s formulas, we deal with complex functions where they take their input as a complex number and the output produced is also a complex number. 

• The conditions that are applied here will be slightly different and stronger than real numbers. Since the complex numbers are two dimensional objects in a plane , their limits could be from any direction.

Cauchy’s integral theorem:

Complex numbers are actually two dimensional numbers that exist on the entire complex plane. It is actually a theorem about the complex functions of Z around a closed curve which is also known as contour integrals.

And so Cauchy’s integral theorem states that, 

Suppose if f(z) is a complex function and C is is a closed curve in a complex plane, when

•  f(z) is holomorphic and inside the closed curve C

• Where C is a simple curve and does not cross anywhere

• And C must have a finite number of corners.

Therefore, Cauchy’s theorem is given by,

                                               cf(z)dz = 0

Where C is a closed curve.

Cauchy’s Integral Formula:

Cauchy’s integral formula states that the values of holomorphic function inside a disk are determined by the values of that function on the boundary of the disk.

And so, if the function is analytic then the closed curve’s value is zero. If it is not analytic then the closed curve value is not equal to zero. This can be further explained as,

If, f : U → C is holomorphic and 𝛾 is a circle contained in U. Then for any ‘a’ value in the disk which is bounded by 𝛾 can be given by,

                         f(a) = 12𝜋i f(z)(z-a)dz. —(1)

Here, 1 / (z-a) can be expanded in terms of power of series, therefore equation(1) can be written as,

                     f(n)(a) = n!n!2𝜋i 𝛾f(z)(z-a)n+1 dz —(2)

Here equation(2) indicates the Cauchy’s Integral formula. And so here the function of f is calculated or formed by differentiating the closed integral values of f(z)with respect to dz. And since f(z) can be written in a power series, the equation is thus re-written in terms of factorials, differentiated with dz.

Difference between Cauchy’s Integral theorem and Cauchy’s Integral formula:

• Here Cauchy’s integral theorem is also based upon generating integral formulas for complex equations.

•  But when it comes to poles which are present inside the loop, it is not possible to derive the equations.

• In the case of Cauchy’s Integral Formula, it can be used to find integral or differential formulas with higher order poles. 

In Between what is Cauchy’s Goursat theorem and Cauchy’s Riemann theorem?

Cauchy’s Goursat theorem:

Cauchy’s Goursat theorem is when a function f is analytic at all points inside and everywhere to closed contour curve C then 

                                              ∫C f(z) dz = 0.

Cauchy’s Riemann theorem:

They are the partial derivatives of u(x,y) and v(x,y) where they are used to compute the derivatives and also to check if the f is a complex derivative.

Uses and application of Cauchy’s Integral formula:

♦ It is used to derive the exact integral formula for cyclotomic polynomials, where cyclotomic polynomials means that they are monic polynomials with integer coefficients that are irreducible over rational numbers.

♦ Therefore if we choose a path in a closed curve C like a or b, the integral prevailing will always have the same value when f(z) is analytic in the concerned region.

♦ In real life, they are used in various fields in mathematics like to solve long complex analysis, discrete mathematics and also in number theory

Also it is used to derive integral formulas for cyclotomic and other classes of polynomials.

Conclusion:

According to Cauchy’s formulas, they are defined in a complex plane and moreover used in complex analysis. Cauchy’s integral formula is used to find or it provides integral formulas for more complex values function with complex variables or holomorphic functions, where holomorphic functions is actually a function with more complex variables and its complex differentiable. It also includes Cauchy’s theorem where it is also used to find derivatives of integrals but without poles. And this is one of the differences between them. Also they have their applications in mathematics where it can be used to find derivatives of calculus, maybe for complex analysis, also used in number theory, discrete mathematics and in solving some cyclotomic and other kinds of polynomials.