{"id":5103,"date":"2021-08-31T11:24:40","date_gmt":"2021-08-31T05:54:40","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5103"},"modified":"2026-02-17T01:08:03","modified_gmt":"2026-02-16T19:38:03","slug":"what-is-cauchys-integral-theorem","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-cauchys-integral-theorem\/","title":{"rendered":"What is Cauchy’s integral theorem?"},"content":{"rendered":"\n

Overview- Cauchy’s integral theorem<\/h2>\n\n\n\n

The integration of a function of a complex variable along an open or close curve in the plane of the complex variables is known as complex integration. Cauchy’s integral theorem<\/a> is the part of complex integration.<\/p>\n\n\n\n

The study of complex integration is very useful in engineering physics and mathematics as well.<\/p>\n\n\n\n

The concept we use to calculate the centre of mass, centre of gravity, mass moment of inertia of vehicles etc.<\/p>\n\n\n\n

We use it in placing a satellite in its orbit to calculate the velocity and trajectory.<\/p>\n\n\n\n

Definition of complex line integral<\/strong><\/h2>\n\n\n\n

In case of a complex function f(z) the path of the definite integral can be along any curve from z = a to z = b.<\/p>\n\n\n\n

In case the initial point and final point coincide so that c is a close curve, then this integral called contour integral<\/a> and is denoted by-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

If f(z) = u(x, y) + iv(x, y), then since dz = dx + i dy<\/p>\n\n\n\n

We have-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

It shows that the evaluation of the line integral of a complex function can be reduced to the evaluation of two line integrals of the real function.<\/p>\n\n\n\n

Properties of line integral<\/h2>\n\n\n\n

Linearity-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Sense reversal-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Partitioning of path-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

ML \u2013 inequality-<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example: Evaluate <\/strong><\/strong> along the path y = x.<\/strong><\/p>\n\n\n\n

Sol.<\/strong><\/p>\n\n\n\n

Along the line y = x,<\/p>\n\n\n\n

dy = dx that dz = dx + i dy <\/p>\n\n\n\n

dz = dx + i dx = (1 + i) dx<\/p>\n\n\n

\n
\"\"\/<\/figure><\/div>\n\n\n

On putting y = x and dz = (1 + i)dx<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Cauchy integral theorem<\/strong><\/h2>\n\n\n\n

A function f(z) is analytic and its derivative f\u2019(z) continuous at all points inside and on a closed curve c, then <\/strong><\/strong>.<\/strong><\/p>\n\n\n\n

Proof:<\/strong> <\/p>\n\n\n\n

Suppose the region is R which is close by curve c and let-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

By using Green\u2019s theorem-<\/p>\n\n\n

\n
\"\"\/<\/figure><\/div>\n\n\n

Replace <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

So that<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Cauchy\u2019s integral formula<\/strong><\/h2>\n\n\n\n

Cauchy\u2019s integral formula is-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Where f(z) is an analytic function within and on a closed curve C, a is any point within C.<\/p>\n\n\n\n

Example-1: Evaluate https:\/\/thechrysaliscapital.com\/<\/a> by using Cauchy\u2019s integral formula.<\/strong><\/p>\n\n\n\n

Here c is the circle |z – 2| = 1\/2<\/strong><\/p>\n\n\n\n

Sol. <\/p>\n\n\n\n

here<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Find its poles by equating the denominator to zero.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

There is one pole inside the circle, z = 2,<\/p>\n\n\n\n

So that-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Now by using Cauchy\u2019s integral formula, we get-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n