Unit-4: Complex Variable – Differentiation (Complete Theory + Formulas + Examples)
Unit-4: Complex Variable – Differentiation
(Complete Formulas + Theory + Solved Examples)
1. Functions of Complex Variable
Let \( z = x + iy \), where \( x, y \in \mathbb{R} \), \( i^2 = -1 \)A complex function: \( w = f(z) = u(x,y) + i v(x,y) \)
where \( u(x,y) \) = real part, \( v(x,y) \) = imaginary part
2. Limit of f(z)
\[ \lim_{z \to z_0} f(z) = w_0 \] if for every \( \epsilon > 0 \), there exists \( \delta > 0 \) such that\( 0 < |z - z_0| < \delta \implies |f(z) - w_0| < \epsilon \)
(Limit must be same along every path)
3. Continuity at z₀
f is continuous at z₀ if: \[ \lim_{z \to z_0} f(z) = f(z_0) \]4. Differentiability & Analytic Function
f is differentiable at z₀ if: \[ f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} \] exists and is same along all paths. f is Analytic (Holomorphic) in a domain if it is differentiable at every point in that domain.5. Cauchy-Riemann Equations (Necessary Condition for Analyticity)
Cartesian Form:
If \( f(z) = u(x,y) + i v(x,y) \) is analytic, then: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]Polar Form: Let \( z = re^{i\theta} \), \( f(z) = U(r,\theta) + i V(r,\theta) \)
\[ \frac{\partial U}{\partial r} = \frac{1}{r} \frac{\partial V}{\partial \theta}, \quad \frac{1}{r} \frac{\partial U}{\partial \theta} = -\frac{\partial V}{\partial r} \] If C-R equations are satisfied + partial derivatives are continuous → f is analytic (Sufficient condition)6. Harmonic Function
A real-valued function φ(x,y) is harmonic if: \[ \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 \] If f(z) = u + iv is analytic → both u and v are harmonicand v is harmonic conjugate of u
7. Method to Check Analyticity & Find Harmonic Conjugate
Example 1: Show that \( f(z) = z^2 \) is analytic. Find conjugate.
\( z = x + iy \), \( f(z) = (x + iy)^2 = x^2 - y^2 + 2ixy \)
\( u = x^2 - y^2 \), \( v = 2xy \)
\( u_x = 2x = v_y \), \( u_y = -2y = -v_x \) → C-R satisfied
\( u_{xx} + u_{yy} = 2 - 2 = 0 \), harmonic
Analytic everywhere. Harmonic conjugate of u is v = 2xy
\( z = x + iy \), \( f(z) = (x + iy)^2 = x^2 - y^2 + 2ixy \)
\( u = x^2 - y^2 \), \( v = 2xy \)
\( u_x = 2x = v_y \), \( u_y = -2y = -v_x \) → C-R satisfied
\( u_{xx} + u_{yy} = 2 - 2 = 0 \), harmonic
Analytic everywhere. Harmonic conjugate of u is v = 2xy
Example 2: Is \( f(z) = e^x \cos y + i e^x \sin y \) analytic?
u = e^x cos y, v = e^x sin y
u_x = e^x cos y = v_y
u_y = -e^x sin y = -v_x → Yes
f(z) = e^z → Entire function
u = e^x cos y, v = e^x sin y
u_x = e^x cos y = v_y
u_y = -e^x sin y = -v_x → Yes
f(z) = e^z → Entire function
8. Milne-Thomson Method (To find f(z) if u or v is given)
If u(x,y) is given and harmonic, treat u as real part of f(z), replace x → z, y → 0 (since on real axis y=0)Then find imaginary part by differentiation.
Example 3 (Milne-Thomson): u = x^3 - 3xy^2 (known harmonic)
Consider f(z) such that Re f(z) = u
Put x = z, y = 0 → u(z,0) = z^3
So f(z) = z^3 + i (something)
Differentiate: f'(z) = 3z^2
Integrate back or use C-R → v = 3x^2 y - y^3
f(z) = z^3 (purely real on real axis)
Consider f(z) such that Re f(z) = u
Put x = z, y = 0 → u(z,0) = z^3
So f(z) = z^3 + i (something)
Differentiate: f'(z) = 3z^2
Integrate back or use C-R → v = 3x^2 y - y^3
f(z) = z^3 (purely real on real axis)
Example 4: Given u = e^x (x cos y - y sin y)
On real axis y=0: u(x,0) = e^x (x ·1 - 0) = x e^x
So Re f(z) |_{y=0} = z e^z
Hence f(z) = z e^z + constant
f(z) = z e^z
On real axis y=0: u(x,0) = e^x (x ·1 - 0) = x e^x
So Re f(z) |_{y=0} = z e^z
Hence f(z) = z e^z + constant
f(z) = z e^z
9. Conformal Mapping
A transformation w = f(z) is conformal at z₀ if:- f is analytic at z₀
- f'(z₀) ≠ 0
10. Bilinear (Mobius) Transformation
General form: \[ w = \frac{az + b}{cz + d} \quad (ad - bc \neq 0) \]Properties of Mobius Transformation
| Property | Description |
|---|---|
| One-to-one (injective) | Different z → different w |
| Onto (surjective) | Covers entire w-plane except possibly one point |
| Circle → Circle/Line | Maps circles and lines to circles or lines |
| Cross-ratio preserved | Fundamental invariant |
| Inverse is also Mobius | w → z = \frac{dw - b}{-cw + a} |
| Three points determine it | Can map any 3 points to any 3 points |
Standard Mobius Transformations
| Type | Form | Maps |
|---|---|---|
| Translation | w = z + b | Shift |
| Magnification + Rotation | w = a z | |a| scaling, arg(a) rotation |
| Inversion + Reflection | w = 1/z | Maps |z|=1 → itself |
| General | w = \frac{z - \alpha}{\ z - \beta} | Maps α→0, β→∞ |
Example 5: Find Mobius transformation that maps z=1→w=0, z=i→w=1, z=-i→w=-1
Standard: maps unit disk to unit disk, so w = e^{iθ} \frac{z - z_0}{\bar{z_0} z - 1}
Here points symmetric → θ=0, z₀=0
w = z (identity)
Standard: maps unit disk to unit disk, so w = e^{iθ} \frac{z - z_0}{\bar{z_0} z - 1}
Here points symmetric → θ=0, z₀=0
w = z (identity)
Example 6: Map upper half-plane Im(z)>0 to unit disk |w|<1
Standard transformation: \( w = \frac{z - i}{z + i} \)
Check: z=0 → w = -i/i = -1? Wait:
z=0 → (0-i)/(0+i) = -i/i = -1? No:
Actually: \( w = \frac{z - i}{z + i} \): z=0 → -i/i = -1 (on circle)
Correct standard: \( w = \frac{z - i}{z + i} \) maps i→0, real axis → unit circle
Yes, Im(z)>0 → |w|<1
Standard transformation: \( w = \frac{z - i}{z + i} \)
Check: z=0 → w = -i/i = -1? Wait:
z=0 → (0-i)/(0+i) = -i/i = -1? No:
Actually: \( w = \frac{z - i}{z + i} \): z=0 → -i/i = -1 (on circle)
Correct standard: \( w = \frac{z - i}{z + i} \) maps i→0, real axis → unit circle
Yes, Im(z)>0 → |w|<1
Summary Table: Analytic Functions
| f(z) | Analytic? | Where? |
|---|---|---|
| zⁿ (n integer ≥0) | Yes | Everywhere |
| eᶻ, sin z, cos z | Yes | Entire |
| 1/z | Yes | z ≠ 0 |
| |z|², \bar{z}, Re(z) | No | Nowhere |
| z \bar{z} | No | Nowhere |