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Cauchy-Riemann Equations

The Cauchy-Riemann equations play a crucial role in examining differentiability and continuity. These equations are essential for the formation of various holomorphic complex functions in mathematics and are used to study analytic functions. The functions involved in the Cauchy-Riemann equations are real-valued and depend on two variables. The equations are expressed as:

• f(x, y) l(x, y)
• df/dx = dv/dy
• df/dy = dv/dx

In these equations, the variables represent components of a complex-valued function, with both real and imaginary parts. The partial derivatives in these equations must satisfy the Cauchy-Riemann conditions. Holomorphic complex differentiable functions are characterized by these equations, making them valuable for solving partial differential equations. Due to their significance, the Cauchy-Riemann equations are frequently encountered in various competitive exams. The property of a complex function being holomorphic means it is differentiable and related to a specific subset of complex functions.

Cauchy-Riemann Equations in Polar Form

The Cauchy-Riemann equations in polar form are a variant of the standard Cauchy-Riemann equations, adapted to polar coordinates. These equations are part of the complex value function system in mathematics, dealing with both real and imaginary parts of the variables. They are especially useful in solving partial differential equations.
Cauchy Riemann equations in polar form are expressed as: fu/fr = 1/r fv/fo and fv/fr = -1 /r fu/fo.
These equations can be transformed into the Cartesian form for complex functions, where the polar form represents complex numbers in a rectangular shape when plotted on a graph. This transformation is vital for understanding the behavior of holomorphic functions in different coordinate systems.

Summary: The Cauchy-Riemann equations in polar form provide a framework for examining complex functions using polar coordinates, allowing for the analysis of partial differential equations in a more versatile manner. This approach is essential for working with complex numbers and functions in various mathematical contexts.

Cauchy-Riemann Equations and Their Applications

The Cauchy-Riemann equations are fundamental in solving problems involving complex numbers and have applications across various mathematical fields. They are used to determine the values of partial derivatives, commonly denoted as $uu$u and $vv$v, which are the real and imaginary parts of a complex-valued function. These equations ensure that partial derivatives satisfy specific conditions, representing holomorphic, or complex-differentiable, functions.

Applications of Cauchy-Riemann Equations

1. Determining Partial Derivatives: The equations help in finding the partial derivatives of complex functions.
2. Computing Derivatives: They are essential for computing the derivatives of complex functions.
3. Holomorphic Functions: They characterize functions that are holomorphic, meaning they are differentiable in the complex plane.

Polar Form of Cauchy-Riemann Equations

The polar form of the Cauchy-Riemann equations is derived from the Cartesian form and is particularly useful when dealing with complex numbers in polar coordinates. The equations in polar form are:

The polar form represents complex numbers in a rectangular shape when plotted on a graph and is also a conjugate function of harmony.

Conclusion

The Cauchy-Riemann equations are crucial in mathematics for analyzing and solving problems involving complex functions. They ensure the proper behavior of partial derivatives and are used in both Cartesian and polar forms. In polar coordinates, they are expressed as: These equations play a vital role in ensuring the differentiability and continuity of complex functions.

Solved Numericals

Q1. The value of k that makes the complex-valued function analytic, where z = x + iy, is _________.
(Answer in integer)
Ans: 1.999 - 2.001
For a function f(z) = u + iv to be analytic, then u and v should obey Cauchy-Riemann equations.
C-R Equations:

Given,
f(Z) = u(x, y) + iv(x, y)
f(Z) = e^-kx cos 2y - ie^-kx sin 2y

Since f(Z) is analytic, it will satisfy Cauchy-Riemann equation,

Q2. If is analytic, then a, b, c, d, e are ______.
Solution: Given:

Now,
Consider u_x = v_y & u_y = -v_x (C-R equations)
⇒ a = 1, 2b = -3e, 2d = 4
& 2b = -12, 4c = e
∴ a = 1, b = - 6, c = 1, d = 2 & e = 4.

Q3. If f(z) = u + iv is an analytic function of z = x + iy and u – v = e^x (cosy - siny), then f(z) in terms of z is 
Solution:
f(z) = u + iv
⇒ i f(z) = - v + i u
⇒ (1 + i) f(z) = (u - v) + i(u + v)
⇒ F(z) = U + iv, where F(z) = (1 + i) f(z)
U = u – v, V = u + v
Now,
Let F(z) be an analytic function

dV = e^x (sin y + cos y) dx + e^z(cosy – siny) dy
∴ dV = d[e^x(siny + cosy)]
Now,
On integrating
V = e^x (siny + cosy) + c₁
F(z) = U + iV = e^x(cosy - siny) + i e^x (siny + cosy) + ic₁
F = e^x(cosy + isiny) + ie^x (cosy + isiny) + ic₁
F(z) = (1 + i) e^{x + iy} + ic₁ = (1 + i)e^z + ic₁
⇒ (1 + i) F(z) = (1 + i) e^z + ic₁

∴ f(z) = e^z + (1 + i) c 

Q4. An analytic function of a complex variable  is defined as f(z) = x²- y² + iψ (x, y), where ψ(x, y) is a real function. The value of the imaginary part of f(Z) at z = (1 + i) is _______ (round off to 2 decimal places).
Ans: 1.99 - 2.01
Solution: f(z) = ϕ + iψ
ϕ = Real part, ψ = Imaginary part
If f(z) is analytic function

Calculation:
Given, f(z) = x² – y² + i ψ (x, y)

ψ = 2 xy
Given, z = 1 + i
Comparing it with z = x + iy, we get:
∴ x = 1, y = 1

$\therefore \left(𝜓\right)(1,1)=2\times 1\times 1=$ψ = 2 when z = 1 + i