**PUZZLE 15**

A two-digit number, read from left to right, is 4.5 times as large as the same number read from right to left. What is the number?

**PUZZLE 16**

Some ducks are marching across a path. There's a duck in front of two ducks, there's a duck behind two ducks, and there's a duck in the middle of two ducks.

What's the least number of ducks that there could have been?

**SOLUTIONS**

**PUZZLE 15 : 18 and 81**

**PUZZLE 16 : 3 Ducks.**
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**Introduction: **

** **The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory.

- Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.

- Laplace transform is a widely used integral transform.

**Laplace transform is just a shortcut for complex calculations**.

** **
** Real Life Applications:**

- The Laplace transform is one of the most important equations in digital signal processing and electronics.

- In Nuclear physics, Laplace transform is used to get the correct form for radioactive decay.

- The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.
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**Geometric interpretation of a complex function. **

If D is the domain of real-valued functions and u(x,y) and v(x,y) then the system of equations **u = u(x,y)** and **v = v(x,y)** describes a **transformation (or mapping)** from the x y - plane into the u v -plane, also called the w-plane.
Therefore, we consider the function **w= f(z) = u(x,y) + i v (x,y) **
to be a transformation (or mapping) from the set D in the z-plane onto the range R in the w-plane.
**Conformal Mapping:**

A function f: C → C is conformal at a point z₀ if and only if it is holomorphic and its derivative is everywhere non-zero on C.

i.e., if f is analytic at z₀ and f’(z₀) ≠ 0

**Isogonal Mapping:**

An isogonal mapping is a transformation w = f (z) that preserves the magnitudes of local angles, but not their orientation.

**Standard Transformations:**

• Translation

- Maps of the form z → z + k, where k є C

• Magnification and rotation

- Maps of the form z → k z , where k є C

• Inversion

- Maps of the form z → 1 / z

Let w = a z, where a ≠ 0

If a = │a│ e^(i α) and, z = │z │ e^(i θ), then

**w = │a│ │z│ e^i(θ + α ) **

The image of z is obtained by rotating the vector z through the angle α and magnifying or contracting the length of z by the factor │a│.

Thus the transformation w = a z is referred to as a** rotation** or **magnification**.

**Example 1:**

Find the image of the region y > 1 under the map w = ( 1 – i ) z

**Solution:**

Let w = u + i v ; z = x + i y

Given w = ( 1 – i ) z
i.e., z = 1/2 ( 1 + i) w [ since ( 1 – i ) ( 1 + i) = 2]

i.e., x + i y = 1/2 ( 1 + i) (u + i v )
i.e., x = (u- v )/2 ; y = (u+v)/2

Hence the region y >1 is mapped on the region u + v > 2 in w –plane.
**Example 2 :**

Determine the region R of the w plane into which the triangular region D enclosed by the lines
x = 0, y = 0, x + y = 3 is transformed under the transformation w = 2z.

**Solution:**
Let w = u +i v; z = x + i y
Given, w =2 z
i.e., u +i v = 2 (x + i y)
i.e., u = 2 x ; v = 2 y
When x = 0, u = 0
**The line x = 0 is transformed into the line u = 0 in the w – plane**.

When y = 0, v = 0
**The line y = 0 is transformed into the line v = 0 in the w – plane.**

When x + y = 3 , we get

u/2 + v/2 = 3

i.e., u + v = 6
**The line x + y = 3 is transformed into the line u + v = 6 in the w – plane.**

**Circle** is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.

**Chord -** ** is a segment whose endpoints lie on a circle**

Secant - is a line that intersects the circle at two points

**Diameter is the longest secant in a circle. **

**Tangent -** is a line in the plane of the circle that intersects the circle at exactly one point

**Sector -The part of a circle enclosed by two radii of a circle and an arc.**

**The cut piece of pizza is minor sector and the remaining is the major sector.**

**Segment -part of a circle bounded by a chord and an arc**

**Semi-circle -** ** is an arc whose endpoints are endpoints of a diameter**