# Program for power of a complex number in O(log n)

Given a complex number of the form x + yi and an integer n, the task is to calculate the value of this complex number raised to the power n.

Examples:

```Input: num = 17 - 12i, n = 3
Output: -2431 + i ( -8676 )

Input: num = 18 - 13i, n = 8
Output: 16976403601 + i ( 56580909840 )
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: This algorithm works in O(log n) time complexity. Let c = a + bi and n is the exponent then,

```power(x, n) {
if(n == 1)
return x
sq = power(x, n/2);
if(n % 2 == 0)
return cmul(sq, sq);
if(n % 2 != 0)
return cmul(x, cmul(sq, sq));
}
```

As complex number has 2 fields one is real and other is complex hence we store it in an array. We use cmul() function to multiply two arrays where cmul() implements the below multiplication.

```If x1 = a + bi and x2 = c + di
then x1 * x2 = a * c - b * d + (b * c + d * a )i
```

As we can see in power() function the same function is called for input decreased by 1/2 times.
T(n) = T(n / 2) + c therefore, T(n) = log n

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `// Function to return the product ` `// of two complex numbers ` `long` `long``* cmul(``long` `long``* sq1, ``long` `long``* sq2) ` `{ ` `    ``long` `long``* ans = ``new` `long` `long``; ` ` `  `    ``// For real part ` `    ``ans = (sq1 * sq2) - (sq1 * sq2); ` ` `  `    ``// For imaginary part ` `    ``ans = (sq1 * sq2) + sq1 * sq2; ` ` `  `    ``return` `ans; ` `} ` ` `  `// Function to return the complex number ` `// raised to the power n ` `long` `long``* power(``long` `long``* x, ``long` `long` `n) ` `{ ` `    ``long` `long``* ans = ``new` `long` `long``; ` `    ``if` `(n == 0) { ` `        ``ans = 0; ` `        ``ans = 0; ` `        ``return` `ans; ` `    ``} ` `    ``if` `(n == 1) ` `        ``return` `x; ` ` `  `    ``// Recursive call for n/2 ` `    ``long` `long``* sq = power(x, n / 2); ` `    ``if` `(n % 2 == 0) ` `        ``return` `cmul(sq, sq); ` `    ``return` `cmul(x, cmul(sq, sq)); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n; ` `    ``long` `long``* x = ``new` `long` `long``; ` ` `  `    ``// Real part of the complex number ` `    ``x = 18; ` ` `  `    ``// Imaginary part of the complex number ` `    ``x = -13; ` `    ``n = 8; ` ` `  `    ``// Calculate and print the result ` `    ``long` `long``* a = power(x, n); ` `    ``cout << a << ``" + i ( "` `<< a << ``" )"` `<< endl; ` ` `  `    ``return` `0; ` `} `

## Python 3

 `# Python3 implementation of the approach ` ` `  `# Function to return the product ` `# of two complex numbers ` `def` `cmul(sq1, sq2): ` ` `  `    ``ans ``=` `[``0``] ``*` `2` ` `  `    ``# For real part ` `    ``ans[``0``] ``=` `((sq1[``0``] ``*` `sq2[``0``]) ``-`  `              ``(sq1[``1``] ``*` `sq2[``1``])) ` ` `  `    ``# For imaginary part ` `    ``ans[``1``] ``=` `((sq1[``1``] ``*` `sq2[``0``]) ``+`  `               ``sq1[``0``] ``*` `sq2[``1``]) ` ` `  `    ``return` `ans ` ` `  `# Function to return the complex  ` `# number raised to the power n ` `def` `power(x, n): ` ` `  `    ``ans ``=` `[``0``] ``*` `2` `    ``if` `(n ``=``=` `0``): ` `        ``ans[``0``] ``=` `0` `        ``ans[``1``] ``=` `0` `        ``return` `ans ` `     `  `    ``if` `(n ``=``=` `1``): ` `        ``return` `x ` ` `  `    ``# Recursive call for n/2 ` `    ``sq ``=` `power(x, n ``/``/` `2``) ` `    ``if` `(n ``%` `2` `=``=` `0``): ` `        ``return` `cmul(sq, sq) ` `    ``return` `cmul(x, cmul(sq, sq)) ` ` `  `# Driver code ` `if` `__name__ ``=``=` `"__main__"``: ` ` `  `    ``x ``=` `[``0``] ``*` `2` ` `  `    ``# Real part of the complex number ` `    ``x[``0``] ``=` `18` ` `  `    ``# Imaginary part of the ` `    ``# complex number ` `    ``x[``1``] ``=` `-``13` `    ``n ``=` `8` ` `  `    ``# Calculate and print the result ` `    ``a ``=` `power(x, n) ` `    ``print``(a[``0``], ``" + i ( "``, a[``1``], ``" )"``) ` ` `  `# This code is contributed by ita_c `

## PHP

 ` `

Output:

```16976403601 + i ( 56580909840 )
```

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