GTU 3140610 Complex Variables and Partial Differential Equations Summer 2025 Question Paper & PDF

Verify that 𝑢(𝑥, 𝑦) = 𝑦³ − 3𝑥²𝑦 is harmonic in the whole complex plane and find its harmonic conjugate 𝑣(𝑥, 𝑦)

Solve 𝜕2𝑢 𝜕𝑥𝜕𝑡 = 𝑒−𝑡 cos 𝑥

Using residue theorem, evaluate ∫ cos 𝜋𝑧² (𝑧−2)(𝑧−1) . 𝑐 𝑑𝑧 counterclockwise around the circle 𝑐: |𝑧| =

Evaluate ∫ 𝑧−2 𝑧(𝑧−1) . 𝑐 𝑑𝑧 where 𝑐 is the circle |𝑧| =

Discuss the convergence of ∑ (6𝑛+1 2𝑛+5)2 (𝑧 − 2𝑖)𝑛 ∞ 𝑛=1 and also find the radius and circle of convergence

Find the Mobius transformation that maps 𝑧1 = 0, 𝑧2 = −𝑖, 𝑧3 = 2𝑖 𝑜𝑛𝑡𝑜 𝑤1 = 5𝑖, 𝑤2 = ∞, 𝑤3 = − 𝑖 respectively. What are the invariant points of this transformation.

Identify the type of singularity of 𝑓(𝑧) = 2−𝑒𝑧 𝑧3

Classify second order homogeneous partial differential equations as elliptic, parabolic or hyperbolic

Solve 𝜕3𝑧 𝜕𝑥3 − 3 𝜕3𝑧 𝜕𝑥² 𝜕𝑦 + 4 𝜕3𝑧 𝜕𝑦3 = 𝑒𝑥+2𝑦

Using the method of separation of variable, solve 𝜕𝑢 𝜕𝑥 = 2 𝜕𝑢 𝜕𝑡 + 𝑢, 𝑢(𝑥, 𝑜) = 6𝑒−3𝑥