Problem 23 - Advanced Engineering Mathematics

Question

\(\int_{i}^{1+i} z e^{z} d z\)

Step-by-Step Solution

Answer

The integral evaluates to \( i e^{1+i} - (i-1) e^i \).

Step 1: Understanding the Integral

We are given the integral \( \int_{i}^{1+i} z e^{z} dz \). The limits of integration are from \( i \) to \( 1+i \), and it involves the function \( z e^{z} \). We aim to find the value of this definite integral.

Step 2: Choosing the Integration Technique

The integral has the form \( \int z e^z \, dz \), which suggests using integration by parts. The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \). We will identify parts of the integral as \( u = z \) and \( dv = e^z \, dz \).

Step 3: Applying Integration by Parts

Using \( u = z \), the derivative is \( du = dz \). For \( dv = e^z \, dz \), the antiderivative is \( v = e^z \). Substituting into the integration by parts formula, \( \int z e^z \, dz = z e^z - \int e^z \, dz \).

Step 4: Solving the Integral

Now, solve \( \int e^z \, dz = e^z \). Substituting back, we have \( z e^z - e^z + C \) as the indefinite integral. Therefore, \( \int z e^z \, dz = z e^z - e^z + C \).

Step 5: Evaluating the Definite Integral

Substitute the limits of integration into the expression obtained: \( \left[ z e^z - e^z \right]_{z=i}^{z=1+i} = \left( (1+i)e^{1+i} - e^{1+i} \right) - \left( i e^i - e^i \right) \).

Step 6: Simplifying the Expression

Calculate each term: \( (1+i)e^{1+i} - e^{1+i} \) simplifies to \( i e^{1+i} \); \( i e^i - e^i \) simplifies to \( (i-1)e^i \). Evaluate: \( i e^{1+i} - (i-1)e^i \).

Step 7: Using Euler's Formula to Simplify Further

Apply Euler's formula \( e^{z} = e^{x+iy} = e^x (\cos y + i \sin y) \) to express these in a more familiar form for further simplification or evaluation.

Step 8: Conclusion: Final Evaluation

After simplifying using Euler's formula and substituting back the values, the final value of the integral is calculated. The detailed calculations and simplifications will show \( i e^{1+i} - (i-1) e^i \).

Key Concepts

Integration by Parts

Integration by parts is a technique commonly used in calculus to integrate the product of two functions. It is particularly useful when one function becomes simpler upon differentiation, and the other is easy to integrate.
This method is derived from the product rule of differentiation. The formula for integration by parts is:

• \[ \int u \, dv = uv - \int v \, du \]

Here, we strategically choose parts of the integral such that:

• \( u \) is the function differentiated (e.g., \( u = z \) in our original exercise)
• \( dv \) is the part integrated (e.g., \( dv = e^z \, dz \) in our case)

This formula effectively helps us transform a complex integral into simpler parts, aiding in the solution of integrals that otherwise prove difficult using basic integration rules.

Euler's Formula

Euler's formula is a foundational formula in complex analysis that connects complex numbers with trigonometry and exponential functions. It is expressed as:

• \[ e^{ix} = \cos x + i \sin x \]

This relationship is crucial when dealing with complex exponentials, like in the original exercise. Euler's formula allows us to express complex numbers through familiar trigonometric functions, which can simplify complex integrals significantly.
For example, when integrating functions involving complex numbers, it's often helpful to break down expressions like \( e^{z} \) by writing them in terms of Euler's formula to separate real and imaginary parts. This approach makes the process of integration more straightforward, especially when evaluating definite integrals over complex paths.

Definite Integral

The concept of a definite integral is central to calculus and analysis. It calculates the accumulation of quantities, represented by the area under the curve of a function between two specified bounds.
In mathematical terms, the definite integral from \( a \) to \( b \) of a function \( f(x) \) is given by:

• \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

where \( F(x) \) is the antiderivative or integral of \( f(x) \).
In the context of complex integration, like our original problem \( \int_{i}^{1+i} z e^{z} dz \), we apply the same principles but along a path in the complex plane. By evaluating the function at the boundaries \( i \) and \( 1+i \), we find the accumulated change between these points. The use of definite integrals is essential for accurate calculations in various areas such as physics, engineering, and probability.

Complex Numbers

Complex numbers extend the concept of one-dimensional numbers to a two-dimensional plane by introducing the imaginary unit \( i \), where \( i^2 = -1 \). A complex number is typically represented as \( z = x + yi \), with \( x \) and \( y \) being real numbers.
These numbers are crucial in many areas of mathematics and engineering because they allow for the evaluation and solution of equations that do not have real solutions.

• Each complex number can be depicted as a point or a vector in a plane, known as the complex plane.
• They are used to represent and solve problems with periodic and oscillatory phenomena, such as waves.

In the context of complex integration, as in our given integral, understanding the behavior and properties of complex numbers is imperative. Complex numbers are key in evaluating integrals over specific paths in the complex plane, utilizing techniques like integration by parts. They also enable the utilization of powerful mathematical tools, like Euler’s formula, to simplify and calculate results effectively.