rm(list = ls())
L=50

t=10
R1=function(a) t*(integrate(function(x) log(x)*dgamma(x,a*t,1),0,L)$v-digamma(a*t)*pgamma(L,a*t,1))
R1=Vectorize(R1, "a")
plot(R1,1,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))


t1=20
R2=function(a) t1*(integrate(function(x) log(x)*dgamma(x,a*t1,1),0,L)$v-digamma(a*t1)*pgamma(L,a*t1,1))
R2=Vectorize(R2, "a")
plot(R2,1,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))

t2=30
R3=function(a) t2*(integrate(function(x) log(x)*dgamma(x,a*t2,1),0,L)$v-digamma(a*t2)*pgamma(L,a*t2,1))
R3=Vectorize(R3, "a")
plot(R3,1,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))

t3=40
R4=function(a) t3*(integrate(function(x) log(x)*dgamma(x,a*t3,1),0,L)$v-digamma(a*t3)*pgamma(L,a*t3,1))
R4=Vectorize(R4, "a")
plot(R4,1,30,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))


par(mfrow=c(2,2))
plot(R1,1,10,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))
legend("bottomright",c(expression("t=10")))
plot(R2,1,10,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))
legend("bottomright",c(expression("t=20")))
plot(R3,1,10,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))
legend("bottomright",c(expression("t=30")))
plot(R4,1,10,type="l",lty=1,xlab=expression(alpha),ylab=expression(varphi(alpha)))
legend("bottomright",c(expression("t=40")))




<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/frontend/web/index.php(18): yii\base\Application-&gt;run()
#11 {main}</pre>