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## Main Question or Discussion Point

I'm doing a bit of studying on calculus of complex functions.

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]

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

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