Which Interpolation Method for Complex Numbers Is Most Accurate?

[sansty]

Hi,
I need to do Interpolation of complex numbers let say
z1=x1+i*y1
and
z2=x2+i*y2
now I have two approaches:
1) interpolate real and imaginary parts separately and have the result

or

2) First change the complex numbers into (R,Theta) co-ordinate and then do the interpolation on R and Theta, and then transform it back to (x,y) co-ordinates...

so which of my appraoch is right, as I have seen different results..
Please reply ASAP
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thanks to @mathman and @hamster143 for reply...

I want to further clear my problem...
I am doing Fourier Transformation of a 2D data, and then in Fourier domain, based on some formula, I am collecting values, and in some cases required values comes in between the spacing of two samples in Fourier domain. So to get those values I need to do interpolation, and for time being, I am using "linear interpolation" .
and my problem is my two approaches gives me two different results, and my prof. wants to know why am I using particular approach, with some proof...

 

[mathman]

If by interpolation, you mean linear interpolation, then the first approach (interpolate in x and y) is correct.

 

[hamster143]

Depends on the reason why you need to interpolate them.