# Experimenting with a sequence of complex numbers in Kotlin

## The challenge

Consider the sequence `S(n, z) = (1 - z)(z + z**2 + z**3 + ... + z**n)`

where `z`

is a complex number and `n`

a positive integer (n > 0).

When `n`

goes to infinity and `z`

has a correct value (ie `z`

is in its domain of convergence `D`

), `S(n, z)`

goes to a finite limit `lim`

depending on `z`

.

Experiment with `S(n, z)`

to guess the domain of convergence `D`

of `S`

and `lim`

value when `z`

is in `D`

.

Then determine the smallest integer `n`

such that `abs(S(n, z) - lim) < eps`

where `eps`

is a given small real number and `abs(Z)`

is the modulus or norm of the complex number Z.

Call `f`

the function `f(z, eps)`

which returns `n`

. If `z`

is such that `S(n, z)`

has no finite limit (when `z`

is outside of `D`

) `f`

will return -1.

#### Examples:

I is a complex number such as I * I = -1 (sometimes written `i`

or `j`

).

`f(0.3 + 0.5 * I, 1e-4) returns 17`

`f(30 + 5 * I, 1e-4) returns -1`

For languages that don’t have complex numbers or “easy” complex numbers, a complex number `z`

is represented by two real numbers `x`

(real part) and `y`

(imaginary part).

`f(0.3, 0.5, 1e-4) returns 17`

`f(30, 5, 1e-4) returns -1`

#### Note:

You pass the tests if `abs(actual - exoected) <= 1`

## The solution in Kotlin

Option 1:

```
package solv
private fun modul(x: Double, y: Double): Double {
if (x != 0.0 || y != 0.0)
return Math.sqrt(x * x + y * y)
else
return 0.0
}
fun f(x: Double, y: Double, eps: Double): Int {
if (modul(x, y) >= 1.0)
return -1
return (Math.log(eps) / Math.log(modul(x, y))).toInt()
}
```

Option 2:

```
package solv
import kotlin.math.*
fun f(x: Double, y: Double, eps: Double): Int {
val m = hypot(x, y)
return if (m < 1) log(eps, m).toInt() else -1
}
```

Option 3:

```
package solv
fun f(x: Double, y: Double, eps: Double): Int {
val res = Math.log(eps) / Math.log(Math.hypot(x, y))
return if (res < 0) -1 else res.toInt()
}
```

## Test cases to validate our solution

```
package solv
import org.junit.Assert.*
import org.junit.Test
import java.util.Random
class solvTest {
private fun dotest(x:Double, y: Double, eps: Double, expect: Int) {
val merr = 1.0
println("Testing " + x + " " + y + " " + eps)
val actual = f(x, y, eps)
println("Actual: " + actual)
println("Expect: " + expect)
val inrange = Math.abs(actual - expect) <= merr
if (inrange == false)
{
println("Expected must be near " + expect + ", got " + actual)
}
println("-")
assertEquals(true, inrange)
}
@Test
fun test1() {
dotest(0.64, 0.75, 1e-12, 1952)
dotest(0.3, 0.5, 1e-4, 17)
dotest(30.0, 50.0, 1e-4, -1)
}
}
```