Questions: Complex Functions and Mappings

Question 1

A student defines f(z) = x² − y² + 2ixy (where z = x + iy) and g(z) = x² − y² + 2iy² as two complex-valued functions. From the perspective of complex analysis, what is the key difference?

• A. f is analytic because its real and imaginary parts satisfy the Cauchy-Riemann equations, while g generally does not
• B. g is analytic because both its components are polynomial in x and y
• C. Both are equally valid analytic functions — complex analysis only requires the output to be a complex number
• D. Neither is analytic because they are written in terms of x and y rather than z

Answer: (A)

f(z) = x² − y² + 2ixy = (x + iy)² = z², which is analytic. For g, u = x² − y² and v = 2y². Checking Cauchy-Riemann: ∂u/∂x = 2x but ∂v/∂y = 4y — these are equal only when x = 2y, not everywhere. So g is not analytic. The crucial point: writing f(z) = u(x,y) + iv(x,y) does NOT make it analytic. Analyticity imposes the Cauchy-Riemann coupling between u and v — the condition that elevates a pair of real functions into a genuinely complex-analytic function.

Question 2

What does it mean geometrically for a complex function to be conformal at a point where its derivative is non-zero?

• A. The function maps the neighborhood of that point to a disk
• B. The function maps circles through that point to other circles
• C. The function preserves angles between any two curves passing through that point
• D. The function preserves the Euclidean distances between nearby points

Answer: (C)

A conformal map preserves angles and their orientation. At a point z₀ where f'(z₀) ≠ 0, any two smooth curves meeting at angle θ will map to curves meeting at the same angle θ. This is a geometric consequence of complex differentiability: the derivative acts locally like multiplication by a complex number, which rotates and scales uniformly in all directions. Conformality fails exactly where f'(z₀) = 0 (critical points), where the derivative's rotation/scaling is degenerate. This angle-preservation property is what makes conformal maps powerful tools in physics and engineering.

Question 3

Writing f(z) = u(x,y) + iv(x,y) shows that any complex function is just a pair of real functions u and v, with no additional constraints between them.

• T. True
• F. False

Answer: False

Any complex-valued function can be written in the form u(x,y) + iv(x,y), but the functions that matter in complex analysis — the analytic (holomorphic) ones — must have u and v satisfy the Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x. This coupling is extremely restrictive: it forces both u and v to be harmonic (satisfying Laplace's equation) and means that specifying u almost entirely determines v (up to a constant). Far from being arbitrary, an analytic function's real and imaginary parts are deeply intertwined.

Question 4

The function f(z) = 1/z maps circles that do not pass through the origin to other circles.

• T. True
• F. False

Answer: True

f(z) = 1/z is a Möbius transformation, and Möbius transformations map circles and lines to circles and lines (where a 'line' is a circle of infinite radius). Specifically, circles not passing through the origin map to circles, while circles through the origin map to lines (since z = 0 maps to infinity). This circle-preserving property (with lines as degenerate circles) is fundamental to the study of Möbius transformations and conformal mapping, and it can be verified algebraically by substituting the general equation of a circle into w = 1/z and simplifying.

Question 5

Why is complex differentiability a far more restrictive condition than real differentiability, even though both are defined as limits of difference quotients? What geometric consequence does this extra restrictiveness have?

Model answer: Real differentiability requires only that the limit of [f(x+h)−f(x)]/h exists as h→0 along the real line. Complex differentiability requires the limit of [f(z+h)−f(z)]/h to exist as h→0 from ANY direction in the complex plane — horizontally, vertically, diagonally, or along any curve. This imposes the Cauchy-Riemann equations as a necessary condition, tightly coupling the real and imaginary parts. The geometric consequence is conformality: because the derivative is a single complex number (not a 2×2 matrix), it acts on directions uniformly — rotating and scaling by the same amount in every direction. This is angle preservation.

In real analysis, a function can be differentiable yet quite 'wild' (e.g., differentiable once but not twice). In complex analysis, once-differentiable (analytic) implies infinitely differentiable, and in fact implies the function equals its Taylor series everywhere in its domain. This extraordinary rigidity — one complex derivative implies all — is entirely due to the Cauchy-Riemann coupling. Geometrically, every analytic function with nonzero derivative is a conformal map: it looks locally like a rotation and stretching, the same in every direction, which is why complex analysis is so useful in fluid dynamics, electrostatics, and potential theory.