#mu1=mu2=0 , sigma1=sigma2=1 , rho=0

g <- function(x, y, a, b, mu1, mu2, sigma1, sigma2, rho) {
  u <- log(exp(a * x) - 1)
  v <- log(exp(b * y) - 1)
  z1 <- (u - mu1) / sigma1
  z2 <- (v - mu2) / sigma2
  exponent <- (-1 / (2 * (1 - rho^2))) * (z1^2 + z2^2 - 2 * rho * z1 * z2)
  density <- (1 / (2 * pi * sigma1 * sigma2 * sqrt(1 - rho^2))) * exp(exponent)
  numerator <- a * b * density
  denominator <- (1 - exp(-a * x)) * (1 - exp(-b * y))
  g_val <- numerator / denominator
  return(g_val)
}
#par(mfrow = c(2, 2), mar = c(6, 6, 2, 2))

# برای a=2 و b=3
x_vals <- seq(0 , 1.5, length.out = 100)
y_vals <- seq(0 , 1.5, length.out = 100)
grid <- expand.grid(x = x_vals, y = y_vals)
z_vals <- mapply(function(x, y) {
  g(x, y, a=2, b=3, mu1=0, mu2=0, sigma1=1, sigma2=1, rho=0)
}, grid$x, grid$y)
z_matrix <- matrix(z_vals, nrow = length(x_vals), ncol = length(y_vals))
persp(x_vals, y_vals, z_matrix,
      theta = 50, phi = 30, expand =0.4,
      cex.axis = 0.65,cex.lab  = 0.95,
      col = "skyblue", border = "black",
      xlab = "x", ylab = "y", zlab = "g(x,y)",
      ticktype = "detailed",
      main = "")

# برای a=0.5 و b=0.5
x_vals <- seq(0 , 4, length.out = 100)
y_vals <- seq(0 , 4, length.out = 100)
grid <- expand.grid(x = x_vals, y = y_vals)
z_vals <- mapply(function(x, y) {
  g(x, y, a=0.5, b=0.5, mu1=0, mu2=0, sigma1=1, sigma2=1, rho=0)
  }, grid$x, grid$y)
z_matrix <- matrix(z_vals, nrow = length(x_vals), ncol = length(y_vals))
persp(x_vals, y_vals, z_matrix,
      theta = 35, phi = 35, expand =0.4,
      cex.axis = 0.65,cex.lab  = 0.95,
      col = "skyblue", border = "black",
      xlab = "x", ylab = "y", zlab = "g(x,y)",
      ticktype = "detailed",
      main = "")

# برای a=2 و b=0.8
x_vals <- seq(0 , 3.5, length.out = 100)
y_vals <- seq(0 , 3.5, length.out = 100)
grid <- expand.grid(x = x_vals, y = y_vals)
z_vals <- mapply(function(x, y) {
  g(x, y, a=2, b=0.8, mu1=0, mu2=0, sigma1=1, sigma2=1, rho=0)
}, grid$x, grid$y)
z_matrix <- matrix(z_vals, nrow = length(x_vals), ncol = length(y_vals))
persp(x_vals, y_vals, z_matrix,
      theta = 33.5, phi = 30,expand =0.4,
      cex.axis = 0.65,cex.lab  = 0.95,
      col = "skyblue", border = "black",
      xlab = "x", ylab = "y", zlab = "g(x,y)",
      ticktype = "detailed",
      main = "")

# برای a=0.5 و b=1.5
x_vals <- seq(0 , 5, length.out = 100)
y_vals <- seq(0 , 5, length.out = 100)
grid <- expand.grid(x = x_vals, y = y_vals)
z_vals <- mapply(function(x, y) {
  g(x, y, a=0.5, b=1.5, mu1=0, mu2=0, sigma1=1, sigma2=1, rho=0)
}, grid$x, grid$y)
z_matrix <- matrix(z_vals, nrow = length(x_vals), ncol = length(y_vals))
persp(x_vals, y_vals, z_matrix,
      theta = 33.5, phi = 30,expand =0.4,
      cex.axis = 0.65,cex.lab  = 0.95,
      col = "skyblue", border = "black",
      xlab = "x", ylab = "y", zlab = "g(x,y)",
      ticktype = "detailed",
      main = "")
<pre>An Error occurred while handling another error:
yii\web\HeadersAlreadySentException: Headers already sent in  on line 0. in /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php:366
Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/ErrorHandler.php(136): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/ErrorHandler.php(135): yii\web\ErrorHandler-&gt;renderException()
#3 [internal function]: yii\base\ErrorHandler-&gt;handleException()
#4 {main}
Previous exception:
yii\web\HeadersAlreadySentException: Headers already sent in  on line 0. in /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php:366
Stack trace:
#0 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Response.php(339): yii\web\Response-&gt;sendHeaders()
#1 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(656): yii\web\Response-&gt;send()
#2 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/models/Files.php(696): yii\base\Application-&gt;end()
#3 /var/www/html/prof-homepages/vendor/faravaghi/yii2-filemanager/controllers/FilesController.php(484): faravaghi\filemanager\models\Files-&gt;getFile()
#4 [internal function]: faravaghi\filemanager\controllers\FilesController-&gt;actionGetFile()
#5 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/InlineAction.php(57): call_user_func_array()
#6 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Controller.php(180): yii\base\InlineAction-&gt;runWithParams()
#7 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Module.php(528): yii\base\Controller-&gt;runAction()
#8 /var/www/html/prof-homepages/vendor/yiisoft/yii2/web/Application.php(103): yii\base\Module-&gt;runAction()
#9 /var/www/html/prof-homepages/vendor/yiisoft/yii2/base/Application.php(386): yii\web\Application-&gt;handleRequest()
#10 /var/www/html/prof-homepages/backend/web/index.php(16): yii\base\Application-&gt;run()
#11 {main}</pre>