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Chapter 2: Problem 15

Find the real and imaginary parts \(u\) and \(v\) of the given complex function \(f\) as functions of \(x\) and \(y\).\(f(z)=e^{2 z+i}\)

Short Answer

Expert verified
The real part is \(u = e^{2x} \cos((2y+1))\) and the imaginary part is \(v = e^{2x} \sin((2y+1))\).

Step by step solution

01

Understand the complex function

The complex function is given by \(f(z) = e^{2z+i}\), where \(z = x + yi\) is a complex number.
02

Express the complex number z

Write the complex number in terms of its components: \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part.
03

Substitute z into the function

Substitute \(z = x + yi\) into the function \(f(z) = e^{2z+i}\) to get \(f(z) = e^{2(x + yi) + i}\). This simplifies to \(f(z) = e^{2x + 2yi + i}\).
04

Simplify the exponent

Combine terms in the exponent: \(2x + 2yi + i = 2x + (2y + 1)i\). Now the expression is \(f(z) = e^{2x + (2y+1)i}\).
05

Use Euler's Formula

Apply Euler's Formula, \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\), to the complex exponent: \(f(z) = e^{2x} (\cos((2y+1)) + i\sin((2y+1)))\).
06

Identify real and imaginary parts

The real part \(u\) of the function is \(e^{2x} \cos((2y+1))\), and the imaginary part \(v\) is \(e^{2x} \sin((2y+1))\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euler's Formula
Euler's Formula is a fundamental principle in complex analysis. It states that any complex exponential function can be expressed as:
  • \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \)
This formula helps us connect complex exponentials with trigonometric functions.
In the original exercise, Euler's Formula converts the complex exponential \( e^{(2y+1)i} \) into \( \cos((2y+1)) + i\sin((2y+1)) \). This conversion allows us to separate the complex function into its real and imaginary components.
Remember:
  • \( \cos(\theta) \) gives the real part
  • \( \sin(\theta) \) gives the imaginary part.
Real and Imaginary Parts
Complex numbers have two distinct components: the real part and the imaginary part. If you have a complex number in the form \( a + bi \), then:
  • \( a \) is the real part
  • \( b \) is the imaginary part.
In our exercise, after applying Euler's Formula, the function \( f(z) = e^{2x} (\cos((2y+1)) + i\sin((2y+1))) \) is transformed. We identify:
  • The real part \( u = e^{2x} \cos((2y+1)) \)
  • The imaginary part \( v = e^{2x} \sin((2y+1)) \)
Understanding real and imaginary parts helps in analyzing and graphing complex functions. You can visualize them on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
This separation is not just theoretical but has practical applications in engineering and physics.
Complex Exponentials
Complex exponentials are fundamental in the study of signals and systems, particularly in engineering fields. They often take the form \( e^{z} \) where \( z \) is a complex number. In these expressions:
  • The real part of the exponent affects the amplitude of the exponential function.
  • The imaginary component affects the oscillation through trigonometric functions.
In the exercise, the complex exponential \( e^{2z+i} \) represents an important concept. By rewriting \( z = x + yi \), we explored how each component influences the function:
  • \( e^{2x} \) determines the decay or growth of the function's amplitude.
  • \( e^{(2y+1)i} \) oscillates due to the trigonometric terms \( \cos((2y+1)) \) and \( \sin((2y+1)) \).
Complex exponentials simplify mathematical operations like differentiation and integration of sinusoidal functions. They enable us to handle oscillatory behaviors efficiently.

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Most popular questions from this chapter

Consider the multiple-valued function \(F(z)=(z-1+i)^{1 / 2}\). (a) What is the branch point of \(F ?\) Explain. (b) Explicitly define two distinct branches of \(f_{1}\) and \(f_{2}\) of \(F .\) In each case, state the branch cut.

Find three sets in the complex plane that map onto the set \(\arg (w)=\pi\) under the mapping \(w=z^{3}\).

Find the image of the given set under the principal square root mapping \(w=z^{1 / 2}\). Represent the mapping by drawing the set and its image.the parabola \(x=\frac{9}{4}-\frac{y^{2}}{9}\)

Groups of Isometries In this project we investigate the relationship between complex analysis and the Euclidean geometry of the Cartesian plane. The Euclidean distance between two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) in the Cartiesian plane is $$ d\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} $$ Of course, if we consider the complex representations \(z_{1}=x_{1}+i y_{1}\) and \(z_{2}=\) \(x_{2}+i y_{2}\) of these points, then the Euclidean distance is given by the modulus $$ d\left(z_{1}, z_{2}\right)=\left|z_{2}-z_{1}\right| $$ A function from the plane to the plane that preserves the Euclidean distance between every pair of points is called a Euclidean isometry of the plane. In particular, a complex mapping \(w=f(z)\) is a Euclidean isometry of the plane if $$ \left|z_{2}-z_{1}\right|=\left|f\left(z_{1}\right)-f\left(z_{2}\right)\right| $$ for every pair of complex numbers \(z_{1}\) and \(z_{2}\).(a) Prove that every linear mapping of the form \(f(z)=a z+b\) where \(|a|=1\) is a Euclidean isometry. A group is an algebraic structure that occurs in many areas of mathematics. A group is a set \(G\) together with a special type of function \(*\) from \(G \times G\) to \(G\). The function \(*\) is called a binary operation on \(G\), and it is customary to use the notation \(a * b\) instead of \(*(a, b)\) to represent a value of \(* .\) We now give the formal definition of a group. A group is a set \(G\) together with a binary operation \(*\) on \(G\), which satisfies the following three properties: (i) for all elements \(a, b\), and \(c\) in \(G, a *(b * c)=(a * b) * c\),(b) Prove that composition of functions is a binary operation on Isom \(_{+}(\mathbf{E})\). That is, prove that if \(f\) and \(g\) are functions in Isom \(_{+}(\mathbf{E})\), then the function \(f \circ g\) defined by \(f \circ g(z)=f(g(z))\) is an element in Isom \(_{+}(\mathbf{E})\). (c) Prove that the set Isom \(_{+}(\mathbf{E})\) with composition satisfies property \((i)\) of a group. (d) Prove that the set \(\operatorname{Isom}_{+}(\mathbf{E})\) with composition satisfies property \((i i)\) of a group. That is, show that there exists a function \(e\) in Isom \(_{+}(\mathbf{E})\) such that \(e \circ f=f \circ e=f\) for all functions \(f\) in Isom \(_{+}(\mathbf{E})\). (e) Prove that the set Isom \(_{+}(\mathbf{E})\) with composition satisfies property \((i i i)\) of a group.

Find two sets in the complex plane that are mapped onto the ray \(\arg (w)=\pi / 2\) by the function \(w=z^{2}\).
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