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The book I'm reading from is "Mathematical Methods For Physicists", and in the course of reading the chapter on 'functions of a complex variable' I have run across the equation

[tex]\int^{z_{2}}_{z_{1}}[/tex]

*f(z) dz*=[tex]\int^{x_{2},y_{2}}_{x_{1},y{2}}[/tex][u(x,y) + iv(x,y)][dx + idy]

So to really understand this I tried proving it to myself, that it's true.

When you substitute u(x,y) + iv(x,y) for f(x) you do get what's up there. so simplify that and you get

= [tex]\int^{x_{2},y_{2}}_{x_{1},y{2}}[/tex][u(x,y)dx-v(x,y)]+i[tex]\int^{x_{2},y_{2}}_{x_{1},y{2}}[/tex][v(x,y)dy + u(x,y)dy)]

which is great, but when I try to integrate f(z) = z I get two different answers.

directly integrating

[tex]\int[/tex] z dz =

[tex]\frac{z^{2}}{2}[/tex]

substitute in x + iy for z

and you get

[tex]\frac{x^{2}-y^{2}}{2}[/tex] + ixy

integrating using the substituted formula

we get

[tex]\frac{x^{2}-y^{2}}{2}[/tex] + i2xy

I can't for the life of me prove to myself that those two equations (the substituted integral and the direct integral) are equal.

Any help would be greatly appreciated. Thanks