Does Correlation Coefficient can give non linear relationship between two data point?

Correlation coefficient provide relationship between two data point or two variable. But, this relationship is linear or non-linear. Correlation coefficient gives values between -1 and +1 which defines negative linear or positive linear relationship between two variable. If its zero then does not mean there is no relationship but it may be non-linear.

Correlation coefficient between x and y is given by below equation

p_xy = Cov(x,y)/s_xs_y

Here Cov(x,y)=E[(x-μ_x)(y-μ_y)] is coveriance. μ is mean of x or y.

Here E is expectation given by E[variable]=(Σ(variable_i)*p(variable_i))/N where N is total variable size.

Here E is sum of product of probability of variable and respective variable divide by total size of variable

Here standarad variable is s_x = [Σ(x_i-μ_x/σ)²]^1/2 s_y= [Σ(x_i-μ_x/σ)²]^1/2

Here sigma is to sum of square of difference of number of x or y from its respective mean and divide it with deviation σ. And after sum its root square to find stardarad variable.

Here standard deviation is σ=E[x²]-[E[x]]²