Problem 3 - Complex Analysis

Question

Write the following functions in the form \(f=u+\mathrm{i} v\) and give explicit formulas for \(u\) and \(v\). (a) \(f(z)=\sin z\), (b) \(f(z)=\cos z\) (c) \(f(z)=\sinh z\), (d) \(f(z)=\cosh z\), (e) \(f(z)=\exp \left(z^{2}\right)\), (f) \(f(z)=z^{3}+z\). Show that in every case the CAUCHY-RIEMANN equations are satisfied (for all \(z \in \mathbb{C})\), and conclude that these functions are analytic in \(\mathbb{C}\).

Step-by-Step Solution

Answer

All functions \(f(z)\) can be expressed as \(u + iv\) with Cauchy-Riemann equations satisfied, thus they are analytic.

Step 1: Express f(z) in terms of x and y for f(z)=sin(z)

To express the function in the form \(f(z) = u(x, y) + iv(x, y)\), we start by considering \(z = x + iy\). For \(f(z) = \, \sin(z)\), use Euler's formula: \(\sin(z) = \sin(x + iy) = \sin x \cosh y + i \cos x \sinh y\). Now, \(u(x, y) = \sin x \cosh y\) and \(v(x, y) = \cos x \sinh y\).

Step 2: Verify Cauchy-Riemann equations for f(z)=sin(z)

The Cauchy-Riemann equations are given by: \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\). Compute the partial derivatives: \(\frac{\partial u}{\partial x} = \cos x \cosh y\), \(\frac{\partial v}{\partial y} = \cos x \cosh y\), \(\frac{\partial u}{\partial y} = \sin x \sinh y\), and \(\frac{\partial v}{\partial x} = -\sin x \sinh y\). The equations are satisfied, so \(f(z)\) is analytic.

Step 3: Express f(z) in terms of x and y for f(z)=cos(z)

Let \(z = x + iy\). Then \(f(z) = \cos(z) = \cos(x+iy) = \cos x \cosh y - i \sin x \sinh y\). Thus, \(u(x, y) = \cos x \cosh y\) and \(v(x, y) = -\sin x \sinh y\).

Step 4: Verify Cauchy-Riemann equations for f(z)=cos(z)

Calculate the derivatives: \(\frac{\partial u}{\partial x} = -\sin x \cosh y\), \(\frac{\partial v}{\partial y} = -\sin x \cosh y\), \(\frac{\partial u}{\partial y} = \cos x \sinh y\), \(\frac{\partial v}{\partial x} = \cos x \sinh y\). Both Cauchy-Riemann equations hold, indicating \(f(z)\) is analytic.

Step 5: Express f(z) in terms of x and y for f(z)=sinh(z)

Use \(z = x + iy\). Then \(f(z) = \sinh(z) = \sinh(x + iy) = \sinh x \cos y + i \cosh x \sin y\). Therefore, \(u(x, y) = \sinh x \cos y\) and \(v(x, y) = \cosh x \sin y\).

Step 6: Verify Cauchy-Riemann equations for f(z)=sinh(z)

Compute the partial derivatives: \(\frac{\partial u}{\partial x} = \cosh x \cos y\), \(\frac{\partial v}{\partial y} = \cosh x \cos y\), \(\frac{\partial u}{\partial y} = -\sinh x \sin y\), \(\frac{\partial v}{\partial x} = \sinh x \sin y\). The Cauchy-Riemann equations are satisfied, confirming that \(f(z)\) is analytic.

Step 7: Express f(z) in terms of x and y for f(z)=cosh(z)

With \(z = x + iy\), \(f(z) = \cosh(z) = \cosh(x + iy) = \cosh x \cos y + i \sinh x \sin y\). Thus, \(u(x, y) = \cosh x \cos y\) and \(v(x, y) = \sinh x \sin y\).

Step 8: Verify Cauchy-Riemann equations for f(z)=cosh(z)

Derivatives: \(\frac{\partial u}{\partial x} = \sinh x \cos y\), \(\frac{\partial v}{\partial y} = \sinh x \cos y\), \(\frac{\partial u}{\partial y} = -\cosh x \sin y\), \(\frac{\partial v}{\partial x} = \cosh x \sin y\). The equations hold and \(f(z)\) is analytic.

Step 9: Express f(z) in terms of x and y for f(z)=exp(z^2)

The function becomes \(f(z) = \exp((x+iy)^2) = \exp(x^2 - y^2 + 2ixy)\). Using Euler's formula, \(f(z) = e^{x^2 - y^2} \cdot (\cos(2xy) + i\sin(2xy))\). Thus, \(u(x, y) = e^{x^2 - y^2} \cos(2xy)\) and \(v(x, y) = e^{x^2 - y^2} \sin(2xy)\).

Step 10: Verify Cauchy-Riemann equations for f(z)=exp(z^2)

Compute \(\frac{\partial u}{\partial x}\), \(\frac{\partial v}{\partial y}\), \(\frac{\partial u}{\partial y}\), and \(\frac{\partial v}{\partial x}\), and check the equations. They are satisfied, proving analyticity.

Step 11: Express f(z) in terms of x and y for f(z)=z^3+z

For \(f(z) = (x+iy)^3 + (x+iy)\), expand using binomial theorem: \(x^3 - 3xy^2 + i(3x^2y - y^3) + x + iy\). Rearrange: \(u(x, y) = x^3 - 3xy^2 + x\) and \(v(x, y) = 3x^2y - y^3 + y\).

Step 12: Verify Cauchy-Riemann equations for f(z)=z^3+z

Differentiate \(u(x, y)\) and \(v(x, y)\) and substitute them into Cauchy-Riemann equations. Since \(\frac{\partial u}{\partial x} = 3x^2 - 3y^2 + 1\), \(\frac{\partial v}{\partial y} = 3x^2 - 3y^2 + 1\), \(\frac{\partial u}{\partial y} = -6xy\), and \(\frac{\partial v}{\partial x} = 6xy\), they verify the equations. Therefore, \(f(z)\) is analytic.

Key Concepts

Analytic Functions

In complex analysis, a function is called *analytic* if it is locally given by a convergent power series. More informally, if you can approximate the function closely by polynomials within a certain region, it’s considered analytic in that region.
Analytic functions are incredibly smooth, meaning they have derivatives of all orders. This property makes them a major focus of study in complex analysis. Being analytic in a domain implies that these functions follow the rules of calculus we’re familiar with from real-variable calculus, like sums, products, and compositions of these functions, remaining analytic.
For example, polynomial functions, exponential functions, and trigonometric functions are analytic everywhere in the complex plane. In the exercise, each function like the sine, cosine, hyperbolic sine, and the polynomial functions are tested for analyticity by examining if they satisfy the Cauchy-Riemann equations. This check assures us that their behavior is well-structured and predictable in the world of complex numbers.

Cauchy-Riemann Equations

The *Cauchy-Riemann equations* are a set of two real partial differential equations. They give necessary and sufficient conditions for a function to be analytic. If a function, expressed as \(f(x + iy) = u(x, y) + iv(x, y)\), satisfies the Cauchy-Riemann equations at a point, it’s analytic at that point. The two equations are:

• \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\)
• \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\)

These equations ensure that a complex function behaves like a real function would—as a smoothly varying map.
In practice, when checking for analyticity, computing these partial derivatives and confirming that they satisfy the Cauchy-Riemann equations shows that the function’s behavior is regular within the complex plane. It’s why functions like \( \sin(z) \), \( \cos(z) \), and polynomial terms like \( z^3 + z \) can be treated confidently within complex analysis for further operations.

Euler's Formula

*Euler's formula* is one of the most beautiful results in mathematics, bridging a connection between complex numbers and trigonometry. The formula states:\[e^{ix} = \cos(x) + i\sin(x)\]
Using this, we can transform complex exponential forms into trigonometric functions, making it a powerful tool in complex analysis. It greatly simplifies tasks such as converting a complex sine or cosine into expressions involving real and imaginary parts, which is what our exercise demands.
This formula emphasizes the deep link between exponential growth (or decay, in the case of hyperbolic functions) and oscillatory behavior, which is seen frequently in physics with waveforms. In our exercise, Euler's formula was used to express trigonometric functions like \( \sin(z) \), cosines, and their hyperbolic counterparts in terms of exponential functions, showing their intrinsic symmetry and helping to check for analyticity.

Complex Exponential Functions

*Complex exponential functions* extend the concept of exponential functions from real numbers to complex numbers. Written generally as \( e^{z} \), where \(z = x + iy\), these functions have fascinating properties in the complex plane.
The complex exponential can be decomposed using Euler's formula, making these functions key in connecting exponential growth with oscillation through sine and cosine functions. When transformed via \( e^{z} = e^x e^{iy} = e^x(\cos(y) + i\sin(y)) \), we can separate the modulus and the argument.
In our exercise context, particularly for functions like \( \exp(z^2) \), the use of complex exponentials lets us break down the function into components that can easily be verified for analyticity using the Cauchy-Riemann conditions. This makes them a cornerstone for not just theoretical explorations, but practical applications in areas like electrical engineering and quantum physics, where complex exponentials are ever-present.