Unit - 1

Complex Numbers and Complex plane

Q1) Explain complex number, Graphing and modulus of complex number and power of I.

A1)

The sum of a real and an imaginary number is a complex number. A complex number is denoted by the letter z and has the form a + ib. Both a and b are genuine numbers in this case. The value 'a' is known as the real component and is indicated by Re(z), while 'b' is known as the imaginary part and is denoted by Im (z). Ib is also known as an imaginary number.

Some of the examples of complex numbers are

Power of i

The letter I often known as the iota, is used to denote the imaginary component of a complex number. Additionally, the iota(i) can be used to find the square root of negative values. We have the value of i2 = -1, and this is used to find the value of √-4 = √i24 = +2i. The value of i2 = 1 is the fundamental aspect of a complex number. Let us try and understand more about the increasing powers of i.

Graphing of Complex Numbers

The real and imaginary parts of a complex number can be thought of as an ordered pair (Re(z), Im(z)) and can be represented as coordinate points in the euclidean plane. The complex plane, or Argand Plane, is the euclidean plane with regard to complex numbers, named after Jean-Robert Argand. The real part - a, with reference to the x-axis, and the imaginary part - ib, with reference to the y-axis, are used to depict the complex number z = a + ib. Let's try to grasp the meaning of two key words related to the argand plane representation of complex numbers. The argument and modulus of a complex number.

Modulus of the Complex Number: The modulus of a complex number is the distance between a point in the argand plane (a, ib) and the complex number. This distance is defined as r = |a2+b2a2+b2| and is measured from the origin (0, 0) to the point (a, ib). Furthermore, this may be deduced from Pythagoras' theorem, in which the hypotenuse is represented by the modulus, the real component is the base, and the imaginary part is the altitude of the right-angled triangle.

Q2) What are properties of complex number?

A2)

The following properties of complex numbers are helpful to better understand complex numbers and also to perform the various arithmetic operations on complex numbers.

Conjugate of a Complex Number: By taking the identical real component of the complex number and altering the imaginary part of the complex number to its additive inverse, the conjugate of the complex number is generated. Conjugate complex numbers are those in which the sum and product of two complex numbers are both real numbers. For a complex number  z = a + ib, its conjugate is ¯zz¯ = a - ib.

The sum of the complex number and its conjugate is z+z  = ( a + ib) + (a - ib) = 2a, and the product of these complex numbers z.z = (a + ib) × (a - ib) = a2 + b2.

Reciprocal of a complex Number: In the process of dividing one complex number by another, the reciprocal of complex numbers comes in handy. The product of one complex number with the reciprocal of another complex number is the process of division of complex numbers. The reciprocal of the complex number z = a + ib is

This also shows that z≠z−1.

Equality of Complex Numbers: Complex numbers are equal in the same way that real numbers are equal. When the real parts of two complex numbers z1=a1+ib1 and z2=a2+ib2 are equal, a1=a2, and the imaginary parts of both complex numbers are equal, b1=b2, they are said to be equal. Also, in the polar form, two complex numbers are equal if and only if they have the same magnitude and their argument (angle) differs by an integral multiple of the magnitude 2π.

Ordering of Complex Numbers: It is impossible to organise complex numbers. Complex numbers, unlike real numbers and other related number systems, cannot be sorted. There is no ordering of complex numbers that is compatible with addition and multiplication, thus they do not have the structure of an ordered field. In an ordered field, the non-trivial sum of squares is 0; however, in a complex integer, the non-trivial sum of squares is i2 + 12 = 0. The magnitude of complex numbers, which is their distance from the origin, can be measured and represented in a two-dimensional argrand plane.

Euler's Formula: As per euler's formula for any real value θ we have eiθ = Cosθ + iSinθ, and it represented the complex number in the coordinate plane where Cosθ is the real part and is represented with respect to the x-axis, Sinθ is the imaginary part that is represented with respect to the y-axis, θ is the angle made with respect to the x-axis and the imaginary line, which is connecting the origin and the complex number. As per Euler's formula and for the functional representation of x and y we have ex + iy = ex(cosy + isiny) = excosy + iexSiny. This decomposes the exponential function into its real and imaginary parts.

Q3) Write about operations on complex numbers?

A3)

The addition, subtraction, multiplication, and division operations that can be done on natural numbers may likewise be performed on complex numbers. The following are the details of the various arithmetic operations on complex numbers.

Addition: Complete numbers are added in the same way as natural numbers are added. The real part is added to the real part, while the imaginary part is added to the imaginary part in complex numbers. The sum of complex numbers z1+z2=(a+c)+i(b+d) given two complex numbers of the form z1=a+id and z2=c+id. All of the properties of addition apply to complex numbers.

• The sum of two complex numbers is also a complex number, according to the Closure Law. The sum of two complex numbers, z1 and z2, is also a complex number.

• Commutative Law: We have z1+z2=z2+z1 for two complex integers z1, z2.

• Associative Law: We have z1+(z2+z3)=(z1+z2)+z3 for the three complex numbers z1,z2,z3.

• Additive Identity: There exists 0 = 0 + i0 for a complex integer z = a + ib, such that z + 0 = 0 + z = 0.

• Additive Inverse: There is a complex number -z = -a -ib such that z + (-z) = (-z) + z = 0 for the complex number z = a + ib. The additive inverse is -z in this case.

Subtraction: The subtraction of complex numbers follows a similar process of subtraction of natural numbers. Here for any two complex numbers, the subtraction is separately performed across the real part and then the subtraction is performed across the imaginary part. For the complex numbers z1 = a + ib, z2=c+id, we have z1−z2 = (a - c) + i(b - d)

Multiplication: Multiplication of complex numbers differs from multiplication of natural numbers in a few ways. We must utilise the formula i2=1 in this case. The product is z1.z2 = (ca - bd) + i(ad + bc) for the two complex numbers z1 = a + ib, z2 = c + id.

The polar form of multiplication for complex numbers differs slightly from the above-mentioned form of multiplication. To get the product of the complex numbers, multiply the absolute values of the two complex numbers and add their arguments.

For the complex numbers z1=r1(Cosθ1+iSinθ1), and z2 = z2=r1(Cosθ2+iSinθ2), the product of the complex numbers is z1.z2=r1.r2(Cos(θ1+θ2)+iSin(θ1+θ2)).

Division: The division of complex numbers makes use of the formula of reciprocal of a complex number. For the two complex numbers z1 = a + ib, z2 = c + id, we have the division as z1z2=(a+ib)×1(c+id)=(a+ib)×(c−id)(c2+d2).

Q4) What are the algebraic identities of complex number?

A4)

For complex numbers, all algebraic identities hold true. Using algebraic identities of complex numbers, the addition and subtraction of complex numbers with exponents of 2 or 3 can be easily solved.

Complex numbers are points in the plane endowed with additional structure. We consider the set R2 = {(x, y): x, yR}, i.e., the set of ordered pairs of real numbers. Two such pairs are equal if their corresponding components coincide:

(x1, y1) = (x2, y2) iff x1 = x2 and y1 = y2.

With two operations - addition and multiplication - defined below, the set R2 becomes the set C of complex numbers. R2 considered as a set of complex numbers C is called Argand diagram or Argand plane or Gauss plane.

Addition

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2).

Multiplication

(x1, y1)(x2, y2) = (x1x2 - y1y2, x1y2 + x2y1).

Addition is defined component wise in a relatively standard way that extends to spaces of higher dimension. Multiplication on the other hand is peculiar to complex numbers. It only has weakened analogues in R4 (quaternions) and R8 (octonions).

Addition and multiplication of complex numbers inherit most of the properties of addition and multiplication of real numbers:

z + w = w + z and zw = wz (Commutativity),

z + (u + v) = (z + u) + v and z(uv) = (zu)v (Associativity),

z(u + v) = zu + zv (Distributive Law).

Q5) Explain the geometry of complex number and the triangle inequality.

A5)

The geometry of complex numbers

We can depict complex numbers as points in the xy-plane because the complex number z = x + iy is described by two quantities, x and y. The complicated plane is what we call it when we do this. The x-axis is referred to as the real axis since x is the real part of z. Similarly, the imaginary axis is the y-axis.

The triangle inequality

The triangle inequality states that the sum of any two legs' lengths is greater than the length of the third leg in a triangle.

The triangle inequality is a statement about complex magnitudes for complex numbers. Precisely: for complex numbers z1, z2

|z1| + |z2|≥|z1 + z2|

With equality only if one of them is 0 or if arg(z1) = arg(z2). This is illustrated in the following figure.

Triangle inequality: |z1| + |z2|≥|z1 + z2|

We get equality only if z1 and z2 are on the same ray from the origin, i.e. they have the same argument.

Q6) Explain   Radius of convergence in complex analysis.

A6)

By changing the argument of a power series with a positive radius of convergence to a complex variable, it can be transformed into a holomorphic function. The following theorem can be used to define the radius of convergence:

The distance from a to the nearest point where cannot be specified in a fashion that makes it holomorphic is equal to the radius of convergence of a power series centred on a point a.

The disc of convergence is the collection of all points whose distance from an is strictly less than the radius of convergence.

Even if the centre and all coefficients are real, the nearest point is in the complex plane, not necessarily on the real line. For instance, consider the function

Has no singularities on the real line, since 1+z2 has no real roots. Its Taylor series about 0 is given by

The root test shows that its radius of convergence is 1. In accordance with this, the function ƒ(z) has singularities at ±i, which are at a distance 1 from 0.

A simple example

The arctangent function of trigonometry can be expanded in a power series:

It is easy to apply the root test in this case to find that the radius of convergence is 1.

Q7) What is continuous functions and properties of continuous functions.

A7)

If there are no abrupt jumps in a function, it is said to be continuous. This is the gist of the definition that follows.

Definition. If the function f(z) is defined on an open disk around z0 and lim f(z) = f(z0)

z→z0

Then we say f is continuous at z0. If f is defined on an open region A then the phrase ‘f is continuous on A’ means that f is continuous at every point in A.

As usual, we can rephrase this in terms of functions of (x,y):

Fact. f(z) = u(x,y) + iv(x,y) is continuous iff u(x,y) and v(x,y) are continuous as functions of two variables.

Example (Some continuous functions)

i. A polynomial

P(z) = a0 + a1z + a2z2 + ... + anzn

Is continuous on the entire plane. Reason: it is clear that each power (x+iy)k is continuous as a function of (x,y).

Ii. The exponential function is continuous on the entire plane. Reason:

ez = ex+iy = ex cos(y) + iex sin(y).

So the both the real and imaginary parts are clearly continuous as a function of (x,y).

Iii. The principal branch Arg(z) is continuous on the plane minus the non-positive real axis. Reason: this is clear and is the reason we defined branch cuts for arg. We have to remove the negative real axis because Arg(z) jumps by 2π when you cross it. We also have to remove z = 0 because Arg(z) is not even defined at 0.

Iv. The principal branch of the function log(z) is continuous on the plane minus the nonpositive real axis. Reason: the principal branch of log has

Log(z) = log(r) + iArg(z).

So the continuity of log(z) follows from the continuity of Arg(z).

Properties of continuous functions

We have the following qualities of continuous functions because continuity is defined in terms of limits.

Suppose f(z) and g(z) are continuous on a region A. Then

• f(z) + g(z) is continuous on A.
• f(z)g(z) is continuous on A.
• f(z)/g(z) is continuous on A except (possibly) at points where g(z) = 0.
• If h is continuous on f(A) then h(f(z)) is continuous on A.

Using these properties we can claim continuity for each of the following functions:

• Ez2
• Cos(z) = (eiz + e−iz)/2
• If P(z) and Q(z) are polynomials then P(z)/Q(z) is continuous except at roots of Q(z).

Q8) Explain Holomorphic Functions and Derivative. What is The Chain Rule for Holomorphic Functions?

A8)

Definition A function f:D →C defined on an open set D in the complex plane is said to be holomorphic on D if the limit

Is defined for all z ∈ D. The value of this limit is denoted by f0(z), or by

, and is referred to as the derivative of the function f at z.

Note that if f:D →C is a holomorphic function defined on an open set C in the complex plane then f is continuous on D. For let z ∈ D. Then

And thus the function f is continuous at z, as required.

Lemma 5.1 A function f:D →C, defined on an open set D in the complex plane, is holomorphic on D if and only if, given any complex number w belonging to D, and given any positive real number ε, there exists some real positive number δ such that |f(z)−f(w)−(z−w)f0(w)|≤ ε|z−w| whenever |z − w| < δ.

Proof The function f has a well-defined derivative f0(w) at a point w of D if and only if

This limit exists if and only if, given any positive real number ε, there exists some positive real number δ such that

Whenever 0 < |z−w| < δ. The required result follows directly on rearranging the above inequality.

Proposition 5.2 Let f:D → C and g:D → C be holomorphic functions defined over an open set D in the complex plane. Then the sum f + g, difference f − g and product f · g of the functions f and g are holomorphic, where (f + g)(z) = f(z) + g(z), (f − g)(z) = f(z) − g(z) and (f · g)(z) = f(z)g(z) for all z ∈ D. Moreover (f + g)0(z) = f0(z) + g0(z), (f − g)0(z) = f0(z) − g0(z) and (f · g)0(z) = f0(z)g(z) + f(z)g0(z) for all z ∈ D.

Proof The results for f + g and f − g follow easily from the fact that the limit of a sum or difference of two complex-valued functions is the sum or difference of the limits of those functions (Proposition 1.11). The limit of a product of complex-valued functions is the product of the limits of those functions, and therefore

As required.

The Chain Rule for Holomorphic Functions

Proposition 5.3 Let f:D → C and g:E → C be holomorphic functions defined over open sets D and E in the complex plane. Suppose that f(D) ⊂ E. Then the composition function g ◦ f:D →C is a holomorphic function on D, and (g ◦ f)0(z) = g0(f(z))f0(z) for all z ∈ D.

Proof Let z0 be a point of D and let w0 = f(z0). Then

For all z ∈ D satisfying z 6= z0, where

g0(w0)   if w = w0,

Now lim G(w) = G(w0), and therefore the   function G is continuous at w0 w→w0

(Lemma 1.12). It follows that the composition function G ◦ f:D → C is continuous at at z0 (Lemma 1.14), and therefore lim G((f(z)) = G(f(z0)) =

z→z0

G(w0) = g0(f(z0)). It follows from standard properties of limits of complexvalued functions (Proposition 1.11) that the limit defining the derivative

(g ◦ f)0(z0) of g ◦ f at z0 exists, and

As required.

Let γ:(a,b) →C be a complex-valued function defined on an open interval (a,b). Then there are real-valued functions α and β defined on (a,b) such that γ(t) = α(t) + iβ(t). For all t ∈ (a,b). We say that the function γ is differentiable on (a,b) if the functions α and β are differentiable, in which case we define

Q9) State power series and convergence of power series.

A9)

Remember that we were able to "simultaneously" evaluate all geometric series to discover that

If |x|<1, and that the series diverges when |x|≥1. At the time, we thought of x as an unspecified constant, but we could just as well think of it as a variable, in which case the series

Is a function, namely, the function k/(1−x), as long as |x|<1. While k/(1−x) is a reasonably easy function to deal with, the more complicated ∑kxn does have its attractions: it appears to be an infinite version of one of the simplest function types—a polynomial. This leads naturally to the questions: Do other functions have representations as series? Is there an advantage to viewing them in this way?

The geometric series has a special feature that makes it unlike a typical polynomial—the coefficients of the powers of x are the same, namely k. We will need to allow more general coefficients if we are to get anything other than the geometric series.

Definition A power series has the form

With the understanding that an may depend on n but not on x.

Convergence of a Power Series

Since the terms in a power series involve a variable x, the series may converge for certain values of x and diverge for other values of x. For a power series centered at x=a the value of the series at x=a is given by c0 Therefore, a power series always converges at its center. Some power series converge only at that value of x. Most power series, however, converge for more than one value of x. In that case, the power series either converges for all real numbers x or converges for all x in a finite interval. For example, the geometric series  converges for all x in the interval  but diverges for all x outside that interval. We now summarize these three possibilities for a general power series.

Convergence of a Power Series

Consider the power series  The series satisfies exactly one of the following properties:

1. The series converges at x=a and diverges for all x≠a
2. The series converges for all real numbers x.
3. There exists a real number R>0 such that the series converges if x-a<R and diverges if x-a>R.  At the values x where x-a=R, the series may converge or diverge.

Q10) Explain Properties of Power Series.

A10)

Combining Power Series

If we have two power series with the same convergence interval, we can add or subtract them to make a new power series with the same convergence interval. To produce a new power series, we can multiply a power series by a power of x or evaluate a power series at xm given a positive integer m. We can use power series representations of other functions to find power series representations for particular functions. For example, since we know the power series representation for  , we can find power series representations for related functions, such as

We state findings for power series addition and subtraction, composition of a power series, and multiplication of a power series by a power of the variable in Combining Power Series. We express the theorem for power series centred at x=0 for clarity. Power series centred at x=a produce similar results.

Multiplication of Power Series

By multiplying power series, we can build new power series. Finding power series representations for functions is made easier by being able to multiply two power series.

We multiply them in the same manner we multiply polynomials. Let's say we wish to multiply something.

And

It appears that the product should satisfy

We state the basic result about multiplying power series in Multiplying Power Series, demonstrating that if

If the series converges on a common interval I, we can multiply it in this fashion, and the new series converges on the interval I as well.

The proof of this theorem is omitted since it is outside the scope of this text and is usually presented in a more advanced course. We will now use this theorem as an example by determining the power series representation for

Using the power series representations for

Differentiating and Integrating Power Series

Consider a power series

Let f be the function defined by this series that converges on some interval I. We'll answer two questions concerning f in this section.

• Is f differentiable, and how do we find the derivative f′ if it is?

• What is the formula for calculating the indefinite integral f(x)dx?

We know that we can evaluate the derivative of a polynomial with a finite number of terms by differentiating each component independently. Similarly, the indefinite integral can be evaluated by integrating each term independently. We illustrate how to achieve the same thing with convergent power series in this example. That is, assuming that

Converges on some interval I, then

And

Converges on I

Q11) Define Integration along the curve.

A11)

Finding the slope of the tangent line to a graph y=f(x) is easy -- just compute f′(x). Likewise, the area under the curve between x=a and x=b is just  But how do we compute slopes and areas with parametrized curves?

The answer is to express everything in terms of the parameter t. By computing derivatives with respect to t and integrals with respect to t, we can compute everything we want.

For slopes, we are looking for dy/dx. This is a limit

Where the limits are as Δt and Δx and Δy all go to zero. As long as we can take the derivatives of x and y, we can compute dy/dx.

Example: Find the slope of the tangent to the unit circle x=cos(t); y=sin(t) at time t=π/6, i.e. at the point (x,y)=(3√2,12).

Solution:

To find the area under a curve between x=a and x=b, we need to compute

Where x(t1)=a and x(t2)=b. To find the area, we need to both compute a derivative as well as an integral.

Q12) Express  in the form A + iB where A and B are real numbers.

A12)

Given

Therefore,, which is the required form A + iB where A = 0 and B = -1.

Q13) Find the modulus of the complex quantity (2 - 3i)(-1 + 7i).

A13)

The given complex quantity is (2 - 3i)(-1 + 7i)

Let z1 = 2 - 3i and z2 = -1 + 7i

Therefore,

And

Therefore, the required modulus of the given complex quantity  

Q14) Find the multiplicative inverse of the complex number z = 4 - 5i.

A14)

The given complex number is z = 4 - 5i.

We know that every non-zero complex number z = x +iy possesses multiplicative inverse given by

Therefore, using the above formula, we get

Therefore, the multiplicative inverse of the complex number z = 4 - 5i is

Q15) Factorize: x2 + y2

A15)

X2 - (-1) y2 = x2 - I2 y2 = (x + iy)(x - iy)

Q16) Find real values of the number a for which a.i is a solution of the polynomial equation

z4 - 2z3 + 7z2 - 4z + 10 = 0.

Then find all roots of this equation

A16)

Since a.i is a solution of the equation, we have

(a.i)4 - 2(a.i)3 + 7(a.i)2 - 4(a.i) + 10 = 0

<=>

a4 + 2.i.a3 - 7a2 - 4.i.a + 10 = 0

<=>

a4 - 7a2 + 10 = 0  and 2a3 - 4a = 0

<=>

a2 = 2

<=>

a = sqrt(2) or a = - sqrt(2)

Now, we know that sqrt(2).i and - sqrt(2).i are roots of z4 - 2z3 + 7z2 - 4z + 10 = 0 .
This means that z4 - 2z3 + 7z2 - 4z + 10 is divisible by (z - sqrt(2).i)(z + sqrt(2).i) = z2 - 2.

The quotient is (z2 - 2 z + 5). The roots of this polynomial are 1 + 2 i and 1 - 2 i.

The four roots of the given equation are

Sqrt(2).i     -sqrt(2).i     1 + 2 i    1 - 2 i.

Q17) For the following power series determine the interval and radius of convergence.

A17)

Okay, let’s start off with the Ratio Test to get our hands on L.

So, we know that the series will converge if,

So, from the previous step we see that the radius of convergence is

Now, let’s start working on the interval of convergence. Let’s break up the inequality we got in Step 2.

To finalize the interval of convergence we need to check the end points of the inequality from Step 4.

Now, we can do a quick Comparison Test on the first series to see that it converges and we can do a quick Alternating Series Test on the second series to see that is also converges.

The interval of convergence is below and for summary purposes the radius of convergence is also shown.

Interval: 

R=

Q18) For the following power series determine the interval and radius of convergence.

A18)

Okay, let’s start off with the Ratio Test to get our hands on L.

So, we know that the series will converge if,

So, from the previous step we see that the radius of convergence is .Now, let’s start working on the interval of convergence. Let’s break up the inequality we got in Step 2.

To finalize the interval of convergence we need to check the end points of the inequality from Step 4.

Now, the first series is an alternating harmonic series which we know converges (or you could just do a quick Alternating Series Test to verify this) and the second series diverges by the p-series test.

The interval of convergence is below and for summary purposes the radius of convergence is also shown.

Interval: 

R=

Q19) Give a power series representation for the derivative of the following function.

A19)

First let’s notice that we can quickly find a power series representation for this function. Here is that work.

Now, we know how to differentiate power series and we know that the derivative of the power series representation of a function is the power series representation of the derivative of the function.

Therefore, 

Remember that to differentiate a power series all we need to do is differentiate the term of the power series with respect to x.