Complex-derivatives.md

+# 3D derivatives:
+## Basic 3D derivative function:
+```c
+int wderiv_derivative_3d_r(int direction, int n, double complex *f, double complex *deriv)
+```
+Function computes n-th derivative with respect to `direction` of 3D complex function $`f(x,y,z)`$: $`\frac{\partial^n f}{\partial t^n}`$.
+ * `direction` - predefined integer corresponding to direction of derivative: `WDERIV_DX` or `WDERIV_DY` or `WDERIV_DZ`
+ * `f` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `deriv` - pointer to output derivative function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+`deriv` can be the same pointer as `f`, then result overwrites input. 
+## First 3D derivative functions:
+```c
+int wderiv_dfdx_3d_r(double complex *f, double complex *dfdx)
+```
+Function computes first derivative with respect to $`x`$ of 3D complex function $`f(x,y,z)`$: $`\frac{\partial f}{\partial x}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `dfdx` - pointer to output derivative function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+`dfdx` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_dfdy_3d_r(double complex *f, double complex *dfdy)
+```
+Function computes first derivative with respect to $`y`$ of 3D complex function $`f(x,y,z)`$: $`\frac{\partial f}{\partial y}`$.   
+ * `f` - pointer to input function which is array of size [nx*ny*nz]
+ * `dfdy` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `return` - error code
+
+`dfdy` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_dfdz_3d_r(double complex *f, double complex *dfdz)
+```
+Function computes first derivative with respect to $`z`$ of 3D complex function $`f(x,y,z)`$: $`\frac{\partial f}{\partial x}`$.   
+ * `f` - pointer to input function which is array of size [nx*ny*nz]
+ * `dfdz` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `return` - error code
+
+`dfdz` can be the same pointer as `f`, then result overwrites input. 
+## Second 3D derivative functions:
+```c
+int wderiv_d2fdx2_3d_r(double complex *f, double complex *d2fdx2)
+```
+Function computes second derivative with respect to $`x`$ of 3D complex function $`f(x,y,z)`$: $`\frac{\partial^2 f}{\partial x^2}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `d2fdx2` - pointer to output derivative function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+`d2fdx2` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_d2fdy2_3d_r(double complex *f, double complex *d2fdy2)
+```
+Function computes second derivative with respect to $`y`$ of 3D complex function $`f(x,y,z)`$: $`\frac{\partial^2 f}{\partial y^2}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `d2fdy2` - pointer to output derivative function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+`d2fdy2` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_d2fdz2_3d_r(double complex *f, double complex *d2fdz2)
+```
+Function computes second derivative with respect to $`z`$ of 3D complex function $`f(x,y,z)`$: $`(\frac{d^2f}{dz^2})`$.   
+ * `f` - pointer to input function which is array of size [nx*ny*nz]
+ * `d2fdz2` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `return` - error code
+
+`d2fdz2` can be the same pointer as `f`, then result overwrites input. 
+## n-th 3D derivative functions:
+```c
+int wderiv_dnfdxn_3d_r(int n, double complex *f, double complex *dnfdxn)
+```
+Function computes n-th derivative with respect to $`x`$ of 3D complex function $`f(x,y,z)`$: $`(\frac{d^nf}{dx^n})`$.   
+ * `f` - pointer to input function which is array of size [nx*ny*nz]
+ * `dnfdxn` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `return` - error code
+
+`dnfdxn` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_dnfdyn_3d_r(int n, double complex *f, double complex *dnfdyn)
+```
+Function computes n-th derivative with respect to $`y`$ of 3D complex function $`f(x,y,z)`$: $`(\frac{d^nf}{dy^n})`$.   
+ * `f` - pointer to input function which is array of size [nx*ny*nz]
+ * `dnfdyn` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `return` - error code
+
+`dnfdyn` can be the same pointer as `f`, then result overwrites input. 
+
+```c
+int wderiv_dnfdzn_3d_r(int n, double complex *f, double complex *dnfdzn)
+```
+Function computes n-th derivative with respect to $`z`$ of 3D complex function $`f(x,y,z)`$: $`(\frac{d^nf}{dz^n})`$.   
+ * `f` - pointer to input function which is array of size [nx*ny*nz]
+ * `dnfdzn` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `return` - error code
+
+`dnfdzn` can be the same pointer as `f`, then result overwrites input. 
+## 3D gradient:
+```c
+int wderiv_gradient_3d_r(double complex *f, double complex *dfdx, double complex *dfdy, double complex *dfdz);
+```
+Function computes gradient of 3D complex function $`f(x,y,z)`$: $`\nabla f = (\frac{df}{dx},\frac{df}{dy},\frac{df}{dz})`$.
+ * `f` - pointer to input function which is array of size [nx*ny*nz]
+ * `dfdx` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `dfdy` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `dfdz` - pointer to output derivative function which is array of size [nx*ny*nz]
+ * `return` - error code
+## 3D gradient squared:
+```c
+int wderiv_gradient2_3d_r(double complex *f, double complex *gradient_square)
+```
+Function computes squared gradient of 3D complex function $`f(x,y,z)`$: $`|\nabla f|^2 = |\frac{df}{dx}|^2 +|\frac{df}{dy}|^2 +|\frac{df}{dz}|^2 `$.   
+ * `f` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `gradient_square` - pointer to output gradient squared function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+`gradient_square` can be the same pointer as `f`, then result overwrites input. 
+## 3D laplacian:
+```c
+int wderiv_laplace_3d_r(double complex *f, double complex *laplace)
+```
+Function computes laplacian of 3D complex function $`f(x,y,z)`$: $`|\Delta f|^2 = \frac{d^2f}{dx^2} +\frac{d^2f}{dy^2} +\frac{d^2f}{dz^2} `$.   
+ * `f` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `laplace` - pointer to output laplacian function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+`laplace` can be the same pointer as `f`, then result overwrites input. 
+ ## 3D divergence:
+```c
+int wderiv_divergence_3d_r(double complex *f, double complex *divergence)
+```
+Function divergence of 3D complex function $`f(x,y,z)`$: $`\nabla \cdot f = \frac{df}{dx} +\frac{df}{dy} +\frac{df}{dz} `$.   
+ * `f` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `divergence` - pointer to output divergence function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+`divergence` can be the same pointer as `f`, then result overwrites input.
+## 3D curl:
+```c
+int wderiv_curl_3d_r(double complex *fx, double complex *fy, double complex *fz, double complex *curl_x, double complex *curl_y, double complex *curl_z);
+```
+Function computes curl of 3D complex vector $`(f_x, f_y, f_z)`$: $`\nabla \times \vec{f} = (\frac{df_y}{dz} - \frac{df_z}{dy}, \frac{df_z}{dx} - \frac{df_x}{dz}, \frac{df_x}{dy} - \frac{df_y}{dx})`$.
+ * `fx` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `fy` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `fz` - pointer to input function which is array of size [nx\*ny\*nz]
+ * `curl_x` - pointer to output $`x`$-component function which is array of size [nx\*ny\*nz]
+ * `curl_y` - pointer to output $`y`$-component function which is array of size [nx\*ny\*nz]
+ * `curl_z` - pointer to output $`z`$-component function which is array of size [nx\*ny\*nz]
+ * `return` - error code
+
+# 2D derivatives:
+## Basic 2D derivative function:
+```c
+int wderiv_derivative_2d_r(int direction, int n, double complex *f, double complex *deriv)
+```
+Function computes n-th derivative with respect to `direction` of 2D complex function $`f(x,y)`$: $`\frac{\partial^n f}{\partial t^n}`$.
+ * `direction` - predefined integer corresponding to direction of derivative: `WDERIV_DX` or `WDERIV_DY`
+ * `f` - pointer to input function which is array of size [nx\*ny]
+ * `deriv` - pointer to output derivative function which is array of size [nx\*ny]
+ * `return` - error code
+
+`deriv` can be the same pointer as `f`, then result overwrites input. 
+## First 2D derivative functions:
+```c
+int wderiv_dfdx_2d_r(double complex *f, double complex *dfdx)
+```
+Function computes first derivative with respect to $`x`$ of 2D complex function $`f(x,y)`$: $`\frac{\partial f}{\partial x}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny]
+ * `dfdx` - pointer to output derivative function which is array of size [nx\*ny]
+ * `return` - error code
+
+`dfdx` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_dfdy_2d_r(double complex *f, double complex *dfdy)
+```
+Function computes first derivative with respect to $`y`$ of 2D complex function $`f(x,y)`$: $`\frac{\partial f}{\partial y}`$.   
+ * `f` - pointer to input function which is array of size [nx*ny]
+ * `dfdy` - pointer to output derivative function which is array of size [nx*ny]
+ * `return` - error code
+
+`dfdy` can be the same pointer as `f`, then result overwrites input. 
+
+## Second 2D derivative functions:
+```c
+int wderiv_d2fdx2_2d_r(double complex *f, double complex *d2fdx2)
+```
+Function computes second derivative with respect to $`x`$ of 2D complex function $`f(x,y)`$: $`\frac{\partial^2 f}{\partial x^2}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny]
+ * `d2fdx2` - pointer to output derivative function which is array of size [nx\*ny]
+ * `return` - error code
+
+`d2fdx2` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_d2fdy2_2d_r(double complex *f, double complex *d2fdy2)
+```
+Function computes second derivative with respect to $`y`$ of 2D complex function $`f(x,y)`$: $`\frac{\partial^2 f}{\partial y^2}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny]
+ * `d2fdy2` - pointer to output derivative function which is array of size [nx\*ny]
+ * `return` - error code
+
+`d2fdy2` can be the same pointer as `f`, then result overwrites input. 
+ 
+## n-th 2D derivative functions:
+```c
+int wderiv_dnfdxn_2d_r(int n, double complex *f, double complex *dnfdxn)
+```
+Function computes n-th derivative with respect to $`x`$ of 2D complex function $`f(x,y)`$: $`(\frac{d^nf}{dx^n})`$.   
+ * `f` - pointer to input function which is array of size [nx*ny]
+ * `dnfdxn` - pointer to output derivative function which is array of size [nx*ny]
+ * `return` - error code
+
+`dnfdxn` can be the same pointer as `f`, then result overwrites input. 
+ 
+```c
+int wderiv_dnfdyn_2d_r(int n, double complex *f, double complex *dnfdyn)
+```
+Function computes n-th derivative with respect to $`y`$ of 2D complex function $`f(x,y)`$: $`(\frac{d^nf}{dy^n})`$.   
+ * `f` - pointer to input function which is array of size [nx*ny]
+ * `dnfdyn` - pointer to output derivative function which is array of size [nx*ny]
+ * `return` - error code
+
+`dnfdyn` can be the same pointer as `f`, then result overwrites input. 
+
+## 2D gradient:
+```c
+int wderiv_gradient_2d_r(double complex *f, double complex *dfdx, double complex *dfdy);
+```
+Function computes gradient of 2D complex function $`f(x,y)`$: $`\nabla f = (\frac{df}{dx},\frac{df}{dy})`$.
+ * `f` - pointer to input function which is array of size [nx*ny]
+ * `dfdx` - pointer to output derivative function which is array of size [nx*ny]
+ * `dfdy` - pointer to output derivative function which is array of size [nx*ny]
+ * `return` - error code
+
+## 2D gradient squared:
+```c
+int wderiv_gradient2_2d_r(double complex *f, double complex *gradient_square)
+```
+Function computes squared gradient of 2D complex function $`f(x,y)`$: $`|\nabla f|^2 = |\frac{df}{dx}|^2 +|\frac{df}{dy}|^2`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny]
+ * `gradient_square` - pointer to output gradient squared function which is array of size [nx\*ny]
+ * `return` - error code
+
+`gradient_square` can be the same pointer as `f`, then result overwrites input. 
+## 2D laplacian:
+```c
+int wderiv_laplace_2d_r(double complex *f, double complex *laplace)
+```
+Function computes laplacian of 2D complex function $`f(x,y)`$: $`|\Delta f|^2 = \frac{d^2f}{dx^2} +\frac{d^2f}{dy^2}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny]
+ * `laplace` - pointer to output laplacian function which is array of size [nx\*ny]
+ * `return` - error code
+
+`laplace` can be the same pointer as `f`, then result overwrites input. 
+
+## 2D divergence:
+```c
+int wderiv_divergence_2d_r(double complex *f, double complex *divergence)
+```
+Function divergence of 2D complex function $`f(x,y)`$: $`\nabla \cdot f = \frac{df}{dx} +\frac{df}{dy}`$.   
+ * `f` - pointer to input function which is array of size [nx\*ny]
+ * `divergence` - pointer to output divergence function which is array of size [nx\*ny]
+ * `return` - error code
+
+`divergence` can be the same pointer as `f`, then result overwrites input.
+
+# 1D derivatives:
+## Basic 1D derivative function:
+```c
+int wderiv_derivative_1d_r(int direction, int n, double complex *f, double complex *deriv)
+```
+Function computes n-th derivative with respect to `direction` of 1D complex function $`f(x)`$: $`\frac{\partial^n f}{\partial t^n}`$.
+ * `direction` - predefined integer corresponding to direction of derivative: `WDERIV_DX`
+ * `f` - pointer to input function which is array of size [nx]
+ * `deriv` - pointer to output derivative function which is array of size [nx]
+ * `return` - error code
+
+`deriv` can be the same pointer as `f`, then result overwrites input. 
+## First 1D derivative functions:
+```c
+int wderiv_dfdx_1d_r(double complex *f, double complex *dfdx)
+```
+Function computes first derivative with respect to $`x`$ of 1D complex function $`f(x)`$: $`\frac{\partial f}{\partial x}`$.   
+ * `f` - pointer to input function which is array of size [nx]
+ * `dfdx` - pointer to output derivative function which is array of size [nx]
+ * `return` - error code
+
+`dfdx` can be the same pointer as `f`, then result overwrites input. 
+
+## Second 1D derivative functions:
+```c
+int wderiv_d2fdx2_1d_r(double complex *f, double complex *d2fdx2)
+```
+Function computes second derivative with respect to $`x`$ of 1D complex function $`f(x)`$: $`\frac{\partial^2 f}{\partial x^2}`$.   
+ * `f` - pointer to input function which is array of size [nx]
+ * `d2fdx2` - pointer to output derivative function which is array of size [nx]
+ * `return` - error code
+
+`d2fdx2` can be the same pointer as `f`, then result overwrites input. 
+ 
+## n-th 1D derivative functions:
+```c
+int wderiv_dnfdxn_1d_r(int n, double complex *f, double complex *dnfdxn)
+```
+Function computes n-th derivative with respect to $`x`$ of 1D complex function $`f(x)`$: $`(\frac{d^nf}{dx^n})`$.   
+ * `f` - pointer to input function which is array of size [nx]
+ * `dnfdxn` - pointer to output derivative function which is array of size [nx]
+ * `return` - error code
+
+`dnfdxn` can be the same pointer as `f`, then result overwrites input. 
+ 
+## 1D gradient:
+```c
+int wderiv_gradient_1d_r(double complex *f, double complex *dfdx);
+```
+Function computes gradient of 1D complex function $`f(x)`$: $`\nabla f = (\frac{df}{dx})`$.
+ * `f` - pointer to input function which is array of size [nx]
+ * `dfdx` - pointer to output derivative function which is array of size [nx]
+ * `return` - error code
+
+## 1D gradient squared:
+```c
+int wderiv_gradient2_1d_r(double complex *f, double complex *gradient_square)
+```
+Function computes squared gradient of 1D complex function $`f(x)`$: $`|\nabla f|^2 = |\frac{df}{dx}|^2`$.   
+ * `f` - pointer to input function which is array of size [nx]
+ * `gradient_square` - pointer to output gradient squared function which is array of size [nx]
+ * `return` - error code
+
+`gradient_square` can be the same pointer as `f`, then result overwrites input. 
+## 1D laplacian:
+```c
+int wderiv_laplace_1d_r(double complex *f, double complex *laplace)
+```
+Function computes laplacian of 1D complex function $`f(x)`$: $`|\Delta f|^2 = \frac{d^2f}{dx^2}`$.   
+ * `f` - pointer to input function which is array of size [nx]
+ * `laplace` - pointer to output laplacian function which is array of size [nx]
+ * `return` - error code
+
+`laplace` can be the same pointer as `f`, then result overwrites input. 
+
+## 1D divergence:
+```c
+int wderiv_divergence_1d_r(double complex *f, double complex *divergence)
+```
+Function divergence of 1D complex function $`f(x)`$: $`\nabla \cdot f = \frac{df}{dx}`$.   
+ * `f` - pointer to input function which is array of size [nx]
+ * `divergence` - pointer to output divergence function which is array of size [nx]
+ * `return` - error code
+
+`divergence` can be the same pointer as `f`, then result overwrites input.