Problem 32 - Advanced Engineering Mathematics

Question

(a) \(\overline{\sin z}=\sin x \cosh y-i \cos x \sinh y=\sin x \cosh (-y)+i \cos x \sinh (-y)=\sin (x-i y)=\sin \bar{z}\) (b) \(\overline{\cos z}=\cos x \cosh y+i \sin x \sinh y=\cos x \cosh (-y)-i \sin x \sinh (-y)=\cos (x-i y)=\cos \bar{z}\)

Step-by-Step Solution

Answer

Both identities \( \overline{\sin z} = \sin \bar{z} \) and \( \overline{\cos z} = \cos \bar{z} \) are verified.

Step 1: Understand the given expression for sine.

The problem provides the expression for the conjugate of the sine of a complex number, \( z = x + iy \). The expression is \( \overline{\sin z} = \sin x \cosh y - i \cos x \sinh y \).

Step 2: Transform the imaginary part.

Recognize that the conjugate of a function changes the sign of the imaginary part. Thus, we replace \( i \) with \( -i \), obtaining \( \sin x \cosh (-y) + i \cos x \sinh (-y) \).

Step 3: Write the expression for \( \sin \bar{z} \).

Find the value of \( \sin(x - iy) \). Using the definition of sine for complex numbers: \( \sin(x - iy) = \sin x \cosh(-y) + i \cos x \sinh(-y) \).

Step 4: Verify identity for sine.

Compare both expressions obtained for \( \overline{\sin z} \) and \( \sin \bar{z} \). Observe that they are identical, verifying \( \overline{\sin z} = \sin \bar{z} \).

Step 5: Understand the given expression for cosine.

The problem provides the expression for the conjugate of the cosine of a complex number, \( z = x + iy \). The expression is \( \overline{\cos z} = \cos x \cosh y + i \sin x \sinh y \).

Step 6: Transform the imaginary part for cosine.

By complex conjugation: \( \cos x \cosh(-y) - i \sin x \sinh(-y) \), changing the sign of the imaginary part accordingly.

Step 7: Write the expression for \( \cos \bar{z} \).

Calculate \( \cos(x - iy) \). Using the definition of cosine for complex numbers: \( \cos(x - iy) = \cos x \cosh(-y) - i \sin x \sinh(-y) \).

Step 8: Verify identity for cosine.

Compare both expressions \( \overline{\cos z} \) and \( \cos \bar{z} \). They are identical, confirming \( \overline{\cos z} = \cos \bar{z} \) holds true.

Key Concepts

Complex Conjugate

In the realm of complex numbers, a complex conjugate is a fundamental concept that helps us understand the behavior of complex functions. A complex number is generally given by the expression \(z = x + iy\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit defined by \(i^2 = -1\). The complex conjugate of \(z\) is denoted by \(\overline{z}\) and is defined as \(\overline{z} = x - iy\). As you can see, the complex conjugate simply involves changing the sign of the imaginary part.

A few important properties of complex conjugates are:

• The conjugate of a real number is the number itself because there is no imaginary part.


• If \(z_1\) and \(z_2\) are complex numbers, \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\).


• The product of a complex number with its conjugate is a real number, i.e., \(z \cdot \overline{z} = x^2 + y^2\).

Understanding how to obtain and work with complex conjugates is essential for simplifying complex expressions and solving problems involving complex variables.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions which are related to hyperbolas instead of circles. These functions appear frequently in the context of complex numbers. The most common hyperbolic functions are the hyperbolic sine \(\sinh(x)\) and hyperbolic cosine \(\cosh(x)\), which are defined by the exponential function as follows:

• Hyperbolic sine: \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).


• Hyperbolic cosine: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).

Some important identities related to these functions which are useful in complex analysis include:

• The square of the hyperbolic cosine minus the square of the hyperbolic sine yields a constant value, known as \(\cosh^2(x) - \sinh^2(x) = 1\).


• The derivatives of these functions follow similar patterns as trigonometric functions: \(\frac{d}{dx}\sinh(x) = \cosh(x)\) and \(\frac{d}{dx}\cosh(x) = \sinh(x)\).

Understanding hyperbolic functions is crucial when dealing with complex expressions involving exponential functions and provides deeper insights into complex number identities.

Complex Sine and Cosine Identities

Complex sine and cosine functions extend their real counterparts to complex arguments and have interesting properties that involve both trigonometric and hyperbolic functions. For a complex number \(z = x + iy\), the complex sine and cosine are defined as:

• Complex sine: \(\sin(z) = \sin(x + iy) = \sin(x)\cosh(y) + i\cos(x)\sinh(y)\).


• Complex cosine: \(\cos(z) = \cos(x + iy) = \cos(x)\cosh(y) - i\sin(x)\sinh(y)\).

These expressions show how the sine and cosine of a complex number combine ordinary trigonometric functions with hyperbolic functions, making them applicable to a wider array of problems. A couple of key identities worth noting include:

• \(\overline{\sin(z)} = \sin(\overline{z})\): The conjugate of the complex sine function equals the sine of the conjugate of \(z\).


• \(\overline{\cos(z)} = \cos(\overline{z})\): Likewise, the conjugate of the complex cosine function equals the cosine of the conjugate of \(z\).

Understanding these identities helps manipulate and integrate complex functions, a common requirement in solving complex analysis problems.