# complex

***

**1. Representing Complex Numbers as Structures**

```c
typedef struct Complex {
    double real;
    double imaginary;
} Complex;
```

**2. Summing Complex Numbers**

```c
Complex sum(Complex a, Complex b) {
    Complex result;
    result.real = a.real + b.real;
    result.imaginary = a.imaginary + b.imaginary;
    return result;
}
```

**3. Subtracting Complex Numbers**

```c
Complex subtract(Complex a, Complex b) {
    Complex result;
    result.real = a.real - b.real;
    result.imaginary = a.imaginary - b.imaginary;
    return result;
}
```

**4. Multiplying Complex Numbers**

```c
Complex multiply(Complex a, Complex b) {
    Complex result;
    result.real = a.real * b.real - a.imaginary * b.imaginary;
    result.imaginary = a.real * b.imaginary + a.imaginary * b.real;
    return result;
}
```

**5. Dividing Complex Numbers**

```c
Complex divide(Complex a, Complex b) {
    Complex result;
    double denominator = b.real * b.real + b.imaginary * b.imaginary;
    result.real = (a.real * b.real + a.imaginary * b.imaginary) / denominator;
    result.imaginary = (a.imaginary * b.real - a.real * b.imaginary) / denominator;
    return result;
}
```

**6. Calculating Complex Conjugate**

```c
Complex conjugate(Complex a) {
    Complex result;
    result.real = a.real;
    result.imaginary = -a.imaginary;
    return result;
}
```

**7. Finding Magnitude of Complex Number**

```c
double magnitude(Complex a) {
    return sqrt(a.real * a.real + a.imaginary * a.imaginary);
}
```

**8. Finding Angle of Complex Number**

```c
double angle(Complex a) {
    return atan2(a.imaginary, a.real);
}
```

**9. Converting Polar to Cartesian Form**

```c
Complex polarToCartesian(double magnitude, double angle) {
    Complex result;
    result.real = magnitude * cos(angle);
    result.imaginary = magnitude * sin(angle);
    return result;
}
```

**10. Converting Cartesian to Polar Form**

```c
Complex cartesianToPolar(double real, double imaginary) {
    Complex result;
    result.real = magnitude(result);
    result.imaginary = angle(result);
    return result;
}
```

**11. Computing Mean of Complex Numbers**

```c
Complex mean(Complex* a, int n) {
    Complex sum = {0, 0};
    for (int i = 0; i < n; i++) {
        sum = sum(sum, a[i]);
    }
    sum.real /= n;
    sum.imaginary /= n;
    return sum;
}
```

**12. Computing Variance of Complex Numbers**

```c
double variance(Complex* a, int n) {
    Complex mean = mean(a, n);
    double sum = 0;
    for (int i = 0; i < n; i++) {
        Complex diff = subtract(a[i], mean);
        sum += magnitude(diff) * magnitude(diff);
    }
    return sum / (n - 1);
}
```

**13. Finding Roots of Complex Quadratic Equation**

```c
void solveQuadratic(Complex a, Complex b, Complex c) {
    Complex discriminant = multiply(b, b) - multiply(multiply(a, c), 4);
    Complex sqrtDiscriminant = sqrt(discriminant);

    Complex root1 = divide(sum(b, sqrtDiscriminant), multiply(a, 2));
    Complex root2 = divide(subtract(b, sqrtDiscriminant), multiply(a, 2));

    printf("Roots: %f + %fi, %f + %fi\n", root1.real, root1.imaginary, root2.real, root2.imaginary);
}
```

**14. Generating Random Complex Number**

```c
Complex randomComplex() {
    Complex result;
    result.real = (double)rand() / (double)RAND_MAX;
    result.imaginary = (double)rand() / (double)RAND_MAX;
    return result;
}
```

**15. Sorting Complex Numbers by Real Part**

```c
int compareReal(const void* a, const void* b) {
    Complex* c1 = (Complex*)a;
    Complex* c2 = (Complex*)b;
    return c1->real - c2->real;
}

void sortReal(Complex* a, int n) {
    qsort(a, n, sizeof(Complex), compareReal);
}
```

**16. Sorting Complex Numbers by Imaginary Part**

```c
int compareImaginary(const void* a, const void* b) {
    Complex* c1 = (Complex*)a;
    Complex* c2 = (Complex*)b;
    return c1->imaginary - c2->imaginary;
}

void sortImaginary(Complex* a, int n) {
    qsort(a, n, sizeof(Complex), compareImaginary);
}
```

**17. Binary Search for Complex Number (Real Part)**

```c
int binarySearchReal(Complex* a, int n, double target) {
    int low = 0, high = n - 1;
    while (low <= high) {
        int mid = (low + high) / 2;
        if (a[mid].real == target) {
            return mid;
        } else if (a[mid].real < target) {
            low = mid + 1;
        } else {
            high = mid - 1;
        }
    }
    return -1;
}
```

**18. Binary Search for Complex Number (Imaginary Part)**

```c
int binarySearchImaginary(Complex* a, int n, double target) {
    int low = 0, high = n - 1;
    while (low <= high) {
        int mid = (low + high) / 2;
        if (a[mid].imaginary == target) {
            return mid;
        } else if (a[mid].imaginary < target) {
            low = mid + 1;
        } else {
            high = mid - 1;
        }
    }
    return -1;
}
```

**19. Finding Minimum and Maximum Complex Numbers**

```c
Complex minComplex(Complex* a, int n) {
    Complex min = a[0];
    for (int i = 1; i < n; i++) {
        if (magnitude(a[i]) < magnitude(min)) {
            min = a[i];
        }
    }
    return min;
}

Complex maxComplex(Complex* a, int n) {
    Complex max = a[0];
    for (int i = 1; i < n; i++) {
        if (magnitude(a[i]) > magnitude(max)) {
            max = a[i];
        }
    }
    return max;
}
```

**20. Checking if Complex Numbers are Equal**

```c
int equalComplex(Complex a, Complex b) {
    return a.real == b.real && a.imaginary == b.imaginary;
}
```

**21. Finding Complex Roots of Unity**

```c
void rootsOfUnity(int n) {
    for (int k = 0; k < n; k++) {
        Complex root = polarToCartesian(1, (2 * M_PI * k) / n);
        printf("%f + %fi\n", root.real, root.imaginary);
    }
}
```

**22. Finding Eigenvalues and Eigenvectors of Complex Matrix**

```c
struct Eigenpair {
    Complex eigenvalue;
    Complex eigenvector;
};

Eigenpair eigenvalueEigenvector(Complex matrix, int n) {
    ... // Eigenvalue and eigenvector computation logic
}
```

**23. Performing Complex Gaussian Elimination**

```c
void gaussianElimination(Complex** matrix, int n) {
    ... // Gaussian elimination logic for complex matrices
}
```

**24. Solving Complex Linear Systems**

```c
void solveLinearSystem(Complex** matrix, Complex* b, int n) {
    ... // Solving complex linear systems logic
}
```

**25. Computing Complex Correlation**

```c
double complexCorrelation(Complex* a, Complex* b, int n) {
    Complex meanA = mean(a, n);
    Complex meanB = mean(b, n);

    Complex numerator = {0, 0};
    for (int i = 0; i < n; i++) {
        Complex subA = subtract(a[i], meanA);
        Complex subB = subtract(b[i], meanB);
        numerator = sum(numerator, multiply(subA, conjugate(subB)));
    }

    Complex denominatorA = {0, 0};
    for (int i = 0; i < n; i++) {
        Complex subA = subtract(a[i], meanA);
        denominatorA = sum(denominatorA, multiply(subA, conjugate(subA)));
    }

    Complex denominatorB = {0, 0};
    for (int i = 0; i < n; i++) {
        Complex subB = subtract(b[i], meanB);
        denominatorB = sum(denominatorB, multiply(subB, conjugate(subB)));
    }

    return divide(numerator, multiply(sqrt(denominatorA), sqrt(denominatorB)));
}
```

**26. Performing Fourier Transform on Complex Data**

```c
void fft(Complex* data, Complex* output, int n) {
    ... // Fast Fourier Transform logic for complex data
}
```

**27. Performing Inverse Fourier Transform on Complex Data**

```c
void ifft(Complex* data, Complex* output, int n) {
    ... // Inverse Fast Fourier Transform logic for complex data
}
```

**28. Computing Complex Integral**

```c
Complex complexIntegral(Complex f(Complex), Complex a, Complex b, int n) {
    Complex result = {0, 0};
    double h = (b.real - a.real) / (n - 1);
    for (int i = 1; i < n; i++) {
        Complex x = {a.real + i * h, a.imaginary + i * h};
        result = sum(result, multiply(h, f(x) * 2));
    }
    return result;
}
```

**29. Generating Complex Mandelbrot Set**

```c
void mandelbrot(Complex c, int maxIterations) {
    Complex z = {0, 0};
    int iteration = 0;
    while (iteration < maxIterations && magnitude(z) <= 2) {
        z = multiply(multiply(z, z), c) + c;
        iteration++;
    }
    if (iteration == maxIterations) {
        printf("#");
    } else {
        printf(".", z.real, z.imaginary);
    }
}
```

**30. Generating Complex Julia Set**

```c
void julia(Complex c, Complex z0, int maxIterations) {
    Complex z = z0;
    int iteration = 0;
    while (iteration < maxIterations && magnitude(z) <= 2) {
        z = multiply(multiply(z, z), c) + c;
        iteration++;
    }
    if (iteration == maxIterations) {
        printf("#");
    } else {
        printf(".", z.real, z.imaginary);
    }
}
```

**31. Simulating Complex Brownian Motion**

```c
Complex brownianMotion(int nSteps) {
    Complex result = {0, 0};
    for (int i = 0; i < nSteps; i++) {
        Complex step = randomComplex();
        result = sum(result, step);
    }
    return result;
}
```

**32. Modeling Complex Harmonic Oscillator**

```c
typedef struct ComplexHarmonicOscillator {
    Complex position;
    Complex velocity;
    Complex omega;
} ComplexHarmonicOscillator;

void updateHarmonicOscillator(ComplexHarmonicOscillator* oscillator, double dt) {
    oscillator->position = sum(oscillator->position, multiply(multiply(oscillator->velocity, oscillator->omega), 0.5 * dt));
    oscillator->velocity = sum(oscillator->velocity, multiply(multiply(oscillator->position, oscillator->omega), -0.5 * dt));
}
```

**33. Representing Complex Quaternions**

```c
typedef struct Quaternion {
    Complex real;
    Complex imaginary;
} Quaternion;
```

**34. Multiplying Quaternions**

```c
Quaternion multiplyQuaternions(Quaternion a, Quaternion b) {
    Quaternion result;
    result.real = sum(multiply(a.real, b.real), multiply(a.imaginary, b.imaginary));
    result.imaginary = sum(multiply(multiply(a.real, b.imaginary), -1), multiply(multiply(a.imaginary, b.real), 1));
    return result;
}
```

**35. Computing Complex Exponential Function**

```c
Complex complexExponential(Complex z) {
    return polarToCartesian(exp(z.real), z.imaginary);
}
```

**36. Computing Complex Logarithmic Function**

```c
Complex complexLogarithm(Complex z) {
    return polarToCartesian(log(magnitude(z)), angle(z));
}
```

**37. Solving Complex Polynomial Equations**

```c
void solvePolynomial(Complex coefficients[], int degree) {
    ... // Complex polynomial equation solving logic
}
```

**38. Computing Complex Square Root**

```c
Complex complexSqrt(Complex z) {
    return polarToCartesian(sqrt(magnitude(z)), angle(z) / 2);
}
```

**39. Computing Complex Power Function**

```c
Complex complexPower(Complex base, Complex exponent) {
    return polarToCartesian(pow(magnitude(base), exponent.real) * exp(-exponent.imaginary * angle(base)), exponent.imaginary * log(magnitude(base)) + exponent.real * angle(base));
}
```

**40. Performing Complex Division with Remainder**

```c
void complexDivisionWithRemainder(Complex a, Complex b, Complex* quotient, Complex* remainder) {
    ... // Complex division with remainder logic
}
```

**41. Generating Random Complex Gaussian Distribution**

```c
Complex gaussianComplex(double meanReal, double meanImaginary, double stdDevReal, double stdDevImaginary) {
    Complex result;
    result.real = stdDevReal * sqrt(-2 * log(rand() / (double)RAND_MAX)) * cos(2 * M_PI * rand() / (double)RAND_MAX) + meanReal;
    result.imaginary = stdDevImaginary * sqrt(-2 * log(rand() / (double)RAND_MAX)) * sin(2 * M_PI * rand() / (double)RAND_MAX) + meanImaginary;
    return result;
}
```

**42. Computing Complex Bessel Function**

```c
Complex complexBesselJ(int order, Complex z) {
    ... // Complex Bessel function computation logic
}
```

**43. Computing Complex Gamma Function**

```c
Complex complexGamma(Complex z) {
    ... // Complex Gamma function computation logic
}
```

**44. Performing Complex Roots of Unity by De Moivre's Theorem**

```c
Complex cis(double angle) {
    return polarToCartesian(1, angle);
}

Complex deMoivresTheorem(Complex z, int n) {
    return multiply(pow(magnitude(z), n - 1), cis(angle(z) * (n - 1)));
}
```

**45. Computing Complex Hypergeometric Function**

```c
Complex complexHypergeometric0F1(Complex a, Complex z) {
    ... // Complex Hypergeometric function computation logic
}
```

**46. Performing Complex Inverse Trig Functions**

```c
Complex complexAcos(Complex z) {
    return complexLogarithm(sum(z, sqrt(subtract(multiply(z, z), 1))));
}

Complex complexAsin(Complex z) {
    return complexAcos(sqrt(subtract(1, multiply(z, z))));
}

Complex complexAtan(Complex z) {
    return divide(complexAcos(z), {0, 1});
}
```

**47. Computing Complex Elliptic Functions**

```c
Complex complexEllipticK(Complex m) {
    ... // Complex Elliptic K function computation logic
}

Complex complexEllipticE(Complex m) {
    ... // Complex Elliptic E function computation logic
}
```

**48. Representing Complex Matrices**

```c
typedef struct ComplexMatrix {
    int rows;
    int cols;
    Complex* data;
} ComplexMatrix;
```

**49. Multiplying Complex Matrices**

```c
ComplexMatrix multiplyComplexMatrices(ComplexMatrix a, ComplexMatrix b) {
    ComplexMatrix result = {a.rows, b.cols, malloc(a.rows * b.cols * sizeof(Complex))};

    for (int i = 0; i < a.rows; i++) {
        for (int j = 0; j < b.cols; j++) {
            Complex sum = {0, 0};
            for (int k = 0; k < a.cols; k++) {
                sum = sum(sum, multiply(a.data[i * a.cols + k], b.data[k * b.cols + j]));
            }
            result.data[i * b.cols + j] = sum;
        }
    }

    return result;
}
```

**50. Solving Complex Linear Systems Using LU Decomposition**

```c
void complexLUSolve(ComplexMatrix a, Complex* b, Complex* x) {
    ... // Complex LU decomposition and solve logic
}
```