Cache’s Integral Theorem and Integral Formula

Introduction

Cauchy’s integral theorem is also known as the Cauchy-Goursat theorem in mathematics. Augustin Louis Cauchy and Edouard Goursat developed this theorem. This theorem is used for the line integrals in the complex planes for the holomorphic functions. Cauchy’s integral theorem developed the residue theorem and Cauchy’s integral formula.

 Discussion

Cauchy integral theorem

Fundamental theorem for the complex line integral

Cauchy’s integral theorem states that if in a simply connected area Ω, f(z) is a function then the contour integral is zero for any closed contour C in that area. In an equation it can be expressed properly that is given below:

“∫C f (z) dz = 0”

If in an open area U the function f (z) is a holomorphic function then and c is a curve in this region from x to y then this equation can be developed:

“∫c f’ (z) dz = f(y) – f(x)”

Here ∫c  f’(z)dz is independent for all in that region when the function f(z) has a single antiderivative value. 

The formulation on simply connected regions

Now consider C as a simple open set is the region U then it can be expressed as U⊆C, where C is a holomorphic function and c: [a,b]. Now for U is a curved area then this equation can be developed that given below:

“∫C f (z) dz = 0”

Here U is expressed as a simply connected area or it can be said that U is an area without any “holes”. 

General Formulation

Now consider C as an open set in the simply connected region U or it can be derived as U⊆C, where C is a function of U. Let c: [a,b] and c is homotopic to a constant curve then this equation can be developed:

“∫C f (z) dz = 0”

3.2 Cauchy integral formula

Let C is a simple closed curve and f (z) is a function and this function is dependent on this curve. Consider C is oriented counterclockwise. Then this equation can be developed for any z inside the C:

“f (z) = (1/2πi ) ∫C (f(Z)/(Z − z))dz”

Now it can be concluded that the values of this function can be expressed in everything inside the simply curved region C.  There can be a little change in this formula that is often used in mathematics. The new formulation of this equation can be developed as:

“f (z) = (1/2πi ) ∫C (f(w)/(w − z))dw”

In this equation, Z is replaced with w. 

If the hypothesis of Cauchy’s integral formula is satisfied with the f(z) and C. Now, for all z inside the open set C this equation can be developed:

“f (z) = (n!/2πi ) ∫C (f(w)/(w − z)n+1)dw”

Here n is equal to 1, 2, 3, 4……………

In this equation, C is a simple closed curve. 

The triangle inequality for integral says that mod (z1 + z2) is less than or equal to the summation of mod z1 and mod z2. In this condition, these two variables are laid on the same ray from the region. It can be said in other ways like the difference between these two variables is equal to or less than the mod of the subtraction of these two variables. 

 Application of Cauchy’s integral theorem

Cauchy’s integral formula is used in several sectors in mathematics such as complex analysis, discrete mathematics, number theory, and combinatorics. Nowadays this formula has been used for the calculation of integral formulas. These integral formulas are related to the coefficient of cyclotomic. This formula has been also used for the derivation of the integral formulas in other classes of polynomials. Cauchy’s integral formula gives the proper values of the analytic functions. These analytics functions in a disk are related to the values on the boundary. 

 Cauchy integral theorem examples

Example 1

Valuate I = ∫C (e2z) / z4 dz where C= mod z = 1

Solution: This problem can be solved easily with the help of Cauchy’s integral formula. Now consider f (z) = e2z. Now that equation can be written as:

  I = ∫C (f (z)) / z4 dz = (2πi) / 3! f’’’(0) = 8πi/3

Example 2

Evaluate ∫C (cos z)/ [z (z2+8)] dz

Solution: Now consider f (z) = (cos z)/ [z (z2+8)]. f (z) is inside the curve and analytic on the curve. The roots of (z2+8) are not inside the curve. As a reason this equation can be written as:  ∫C (cos z) (z2+8)/z dz = ∫C f (z)/z dz = 2πi f (0) = 2πi (1/8) = πi/4 

 Conclusion

It can be concluded that Cauchy’s integral theorem and formula have a great impact on several mathematical aspects nowadays. This theorem can help in complex analysis, number theory, and discrete mathematics widely. In recent days this theorem and formula can be used to derive the integral formulas for the coefficient of cyclomatic. Also, this formula can be used for the derivation of the integral formulas in other classes of polynomials.